research article the impact of the strategic advertising...

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 534605 16 pageshttpdxdoiorg1011552013534605

Research ArticleThe Impact of the Strategic Advertising on Luxury FashionBrands with Social Influences

Jin-Hui Zheng1 Bin Shen1 Pui-Sze Chow1 and Chun-Hung Chiu2

1 Business Division Institute of Textiles and Clothing The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong2 Sun Yat-sen Business School Sun Yat-sen University Guangzhou 510275 China

Correspondence should be addressed to Bin Shen binshenjerrygmailcom

Received 6 December 2012 Accepted 23 January 2013

Academic Editor Tsan-Ming Choi

Copyright copy 2013 Jin-Hui Zheng et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

It is well known that purchase of luxury fashion brands is strongly influenced by social needs such as the need for uniqueness and theneed of conformity The existence of these two competing social needs separates customers into two groups who exhibit differentbuying behaviors This paper concerns the impacts of such social influences between different consumer groups on pricing andadvertising strategies of luxury fashion brands with penalty of insufficient advertisingWe start by considering different advertisingallocation strategies and derive the corresponding local optimal pricing and advertising allocation policies through which theglobal optimal policy that maximizes the companyrsquos profit can be obtained Important insights on strategic advertising for luxuryfashion brands are discussed

1 Introduction

With the spectacular growth of consumption in its marketluxury fashion has become an area of growing interest topractitioners as well as academicians Due to its high-pricepositioning luxury fashion was used to be out of reach ofmass consumption however luxury fashion brands are nowdeveloping strategies to manage broader diversity of con-sumer preferences for increasing profits Luxury fashionbrands such as Chanel Christian Dior Gucci and LouisVuitton have been employing a variety of branding strategiesto tackle the problem of being out of reach of mass con-sumption particularly in Asian countries such as ChinaJapan and Korea (notice that for many luxury fashion brandsconsumers from China are critical [1 2] and their conspicu-ous behaviors are very influential for luxury fashion brandsrsquooperations)

In addition to various branding strategies such as brandextensions [3] and strategic pricing [4] advertising is deemedan effective strategy for luxury fashion brands to developand maintain huge demand and consumption Apart fromthe physical value (quality) their products provide luxuryfashion brands highly emphasize the symbolic value (pres-tige) their image project As such advertising is a commontool that luxury brands rely on in building brand salience

Advertising influences not only the ldquoimmediaterdquo purchasebut also the long-term brand equity of the luxury brandAdvertising along with personal experience is an unde-niable force in creating brand equity [5] One mechanismof creating brand equity via advertising is by creating andenhancing brand image

Meanwhile it is well known that consumption is stronglyinfluenced by social needs such as prestige and self-image[6 7] For luxury fashion brands consumers are influencedby two competing social needs the need for uniqueness andthe countervailing need for conformity in purchase [8 9]Accordingly a luxury fashion brand needs to reconcile thepotential tradeoffs between exclusivity and accessibility ontarget advertising groups On the one hand exclusivitymakesa brand out of ordinary as well as projects a positive brandassociation of user profiles on the other hand accessibilityprovides sufficient sales and profits [10 11]

Recognition of the social need for uniqueness and thatfor conformity also implies the existence of two types of con-sumers namely the elites and the majority masses The elitesare usually more fashion conscious and would like to distin-guish themselves from themasses in consumption whilst themasses seek to emulate the choices of the elites (see [12ndash14] forthe details) In this paper following Zheng et al [14] we referto the elite consumers and the majority masses as the leader

2 Mathematical Problems in Engineering

group (LG) and the follower group (FG) respectively Theformer group usually plays a leadership role in the fashiontrend whereas the latter one is usually following the purchaseof LG

In light of the different buying behaviors between the twogroups of consumers luxury fashion brands must target eachof them effectively and efficiently with proper allocation thelimited advertising budget in order to obtain its best payoffTake the classic high-end British fashion brand Burberryas an example Burberry holds its own fashion shows inMilan every year to attract its elite customers The brand alsostrengthens the social interaction with its mass customersit is now the leading luxury fashion brand on Facebookwith over one million fans In short Burberry utilizes thesocial influence for brand development establishing a leadingpresence across social media platforms and creating newcommunities of interest

Zheng et al [14] explore the impact of social influences onthe optimal pricing and advertising allocation strategies of aluxury fashion brand in the presence of LG and FG whichhave social influences on each other In this paper we extendZheng et al [14] with more realistic considerations To bespecific we investigate how a luxury fashion brand allocatesthe limited advertising expense strategically to maximize itsprofit Similar to Zheng et al [14] we consider a retail brandowner that sells a luxury fashion product to the market thatconsists of two groups of consumers (LG and FG) who havesocial influence on each other We consider the scenario thatthere is a certain ldquobasic amountrdquo of advertising effort for eachmarket segment (FG LG) in order to maintain the brandstrength otherwise the brand would suffer in the long rundue to loss of goodwill from a group of consumers To reflectthat a brand has concerns on its budget allocation for sus-taining the respective brandrsquos rolepositioning in the marketwith respect to both groups of consumers we consider thatthere is a penalty cost in the form of a linear loss function forany advertising effort lower than the basic amount for eachmarket segment

2 Literature Review

Inmarketing science and operationsmanagement there are aconsiderable amount of studies that examine optimization ofanalytical models with pricing andor advertising decisionsunder different settings To take into consideration of theeffect of advertising Krahmer [15] considers a model withadvertising that informs the public of brand names and cre-ates the possibility of conspicuous consumption by renderingbrands as a signaling device The author shows that advertis-ing increases consumersrsquo willingness to pay which providesa foundation based on optimization behavior for persuasiveapproaches to advertising Grosset andViscolani [16] proposea model of a firm that advertises a product in a homoge-neous market where a constant exogenous interference ispresent They reveal that the optimal policy takes one of twoforms either a positive and constant advertising effort or adecreasing effort starting from a positive level and eventuallyreaching the zero value at a finite exit time Ghosh and Stock[17] use a model of informative advertising to study the

effect of penetration on competing advertisersrsquo strategies andprofits Conditions under which an increase in penetrationcounter intuitively leads firms to increase advertising levelsand enjoy higher profits are identified In Zheng et al [14] theauthors consider that advertising has a direct impact on socialneeds but lack of considering the impact of the penalty of theinsufficient advertising on the target groups It is importantto consider the loss from the insufficient advertising forthe target group because in practices low advertising efforton the market might have a negative impact on marketdemand

It is well known that advertising on target groups canincrease the grouprsquos consumption In practice most of thetime advertising is targeted [18 19] Anand and Shachar [18]study a model in which firms can target their advertisementsto particular groups of consumers and advertising is noisyThey consider the case that a particular product has a betterfit with the tastes of some consumers than those of the oth-ers and consumer-utility depends on the resulting ldquomatchrdquobetween product attributes and their tastes Raghavan andIyer [19] examine advertising strategy when competing firmscan target their advertising effort to different groups of con-sumers within a market With targeted advertising they findthat firms advertise more to consumers who have a strongpreference for their product than to comparison shopperswho can be attracted to the competition They argue thatadvertising less on comparison shoppers can be seen as away for firms to endogenously increase differentiation in themarket Interestingly they also find that target advertisingleads to higher profits regardless of the firmsrsquo ability to settargeted prices

Our study is related to the work in social factors and con-spicuous consumption [15 20 21] Focusing on the conspic-uous products Krahmer [15] indicates that brands are con-sumed for image reasons and advertising creates a brandrsquosimage He argues that advertising informs the public of brandnames and creates the possibility of conspicuous consump-tion In a price-competition framework he shows that adver-tising increases consumersrsquo willingness to pay and thus pro-vides a foundation for determining the optimal advertisingstrategyMoreover he finds that an incumbentmight strategi-cally overinvest in advertising to deter entry and competitionmight be socially undesirable McClure and Kumcu [20]formalize the relationship between the optimal pricequantitycombination and the thoroughness of conspicuous productpromotionsThey reveal that iterating towards the profitmax-imizing thoroughness of product promotion will lead to abackward bending pricequantity locus In this paper westudy an optimal control problem in the fashion apparelindustryWe establish the studywhich analytically shows howa firm allocates the target advertising strategically on the lux-ury fashion brands when facing the market demand isaffected by consumer desire for exclusivity and conformityrespectively

3 The Model

We consider the scenario that a company sells a fashionproduct to the market with two groups of customers namely

Mathematical Problems in Engineering 3

the leader-group (LG) customers and the follower-group(FG) customers Similar to Zheng et al [14] the demands ofthese two groups are interdependent a higher demand of LGinduces a higher demand of FG but a higher demand of FGimplies a lower demand of LGThe unit product cost is 119888 andthe unit retail price is 119901 where 119901 gt 119888 gt 0 We considerthe case that 119888 is exogenous whilst 119901 is decided by thecompany Both groups are price sensitive and demands ofboth groups are strictly decreasing in 119901 (Chiu et al [22])In order to increase the sales volume of the product thecompany implements advertising campaigns on LG and FGrespectively and the total advertising resource and effort ofthe company spent on the advertising campaigns are denotedby 119890 Let 120582 be the proportion of the effort to be spent on LGwhere 120582 isin [0 1]Then (1minus120582) is the proportion of the effort tobe spent on FG The advertising cost function 119862(119890) is strictlyincreasing in 119890 and the marginal cost of the total advertisingeffort is strictly increasing in 119890 that is 119889119862(119890)119889119890 gt 0 and1198892119862(119890)119889119890

2gt 0 In order to have closed-form solutions to

generate more analytical insights we consider 119862(119890) = ℎ1198902

where ℎ gt 0 We denote the strategy of company withadvertising and pricing by 120596 = 119890 120582 119901 We only focus onthe set of finite 120596 that is 120596 isin Ω = 0 le 119890 lt +infin 0 le 120582 le 1

119888 lt 119901 lt +infin To reflect the interdependency of the demandsof different customer groups in the luxury fashionmarket weadopt the following additive demand functions of LG and FGrespectively

119863119871(120596) = [119881

119871(120596)]+

119863119865(120596) = [119881

119865(120596)]+

(1)

where 119881119871(120596) = 119909

119871+ 119886120582119890 minus 119887119863

119865(120596) minus 119892119901 119881

119865(120596) = 119909

119865+ 119886(1 minus

120582)119890 + 120573119863119871(120596) minus 120574119901 [119884]+ = max0 119884 and 119909

119871 119909119865 119886 120572 119887

120573 119892 and 120574 are all nonnegativeDifferent from Zheng et al [14] we further consider that

certain amount of advertising is needed for each marketsegment in order to maintain the brand strength otherwisethe brand would suffer in the long run due to loss of goodwillfrom a group of consumers We hence consider the followinglinear loss functions (Marinelli [23]) for insufficient advertis-ing for LG and FG Λ

119871(119890) = 119898[119879 minus 120582119890]

+ and Λ119865(119890) = 120583[120591 minus

(1 minus 120582)119890]+ respectively where 119879 ge 0 and 120591 ge 0 are the

minimumadvertising effortresources for LG andFG respec-tively and 119898 ge 0 and 120583 ge 0 are the marginal losses dueto insufficient advertising effortresources that are assignedfor LG and FG respectively Accordingly the profit of thecompany is

120587119871119871(120596) = (119901 minus 119888)119863 (120596) minus 119862 (119890) minus Λ

119871(119890) minus Λ

119865(119890)

= (119901 minus 119888)119863 (120596) minus ℎ1198902minus 119898[119879 minus 120582119890]

+

minus 120583[120591 minus (1 minus 120582) 119890]+

(2)

In this section our objective is to derive the optimaladvertising andpricing strategy for the social influencemodelwith linear loss function for insufficient advertising Mathe-matically we consider the following optimization model

max120596isinΩ


119871(119890) minus Λ

119865(119890)

(P-SILL)

whereΩ = 119890 ge 0 0 le 120582 le 1 119901 gt 119888 (SILL stands for SocialInfluence with Linear (L)oss penalty for insufficient advertis-ing)

Denote the optimal solution of (P-SILL) by 120596lowast = 119890lowast

120582lowast 119901lowast Because of the presence of the penalty functions

of insufficient advertising there exist some situations that120587119871119871(120596lowast) lt 0 that is the company is not profitable In this

paper we only consider the situations that the company isprofitable that is there exist some feasible 120596 that 120587

119871119871(120596) gt 0

Similar to Zheng et al [14] we have the following assumption

Assumption 1 119909119871gt 119892119888 and 119909

119865gt 120574119888

We employ Assumption 1 to make sure that there arepositive product demands in bothmarket segments when theproduct is sold at production cost and the social influencesare not considered

Notice that since Λ119871(119890) and Λ

119865(119890) are nondifferentiable

at 120582119890 = 119879 and (1 minus 120582)119890 = 120591 respectively 120587119871119871(120596) is non-dif-

ferentiable at 119890 = 119879120582 and 119890 = 120591(1 minus 120582)To deal with the fact that 120587

119871119871(120596) is non-differentiable at

some points we consider the following four exclusive strate-gies in determining the optimal advertising and pricingpolicy for the company

Strategy 1 Advertising effort assigned to both market seg-ments are sufficient that is 120582119890 ge 119879 and (1 minus 120582)119890 ge 120591 In thiscase we have 119890 ge 119879 + 120591

Strategy 2 Advertising effort assigned to LG is sufficient butthat to FG is insufficient that is 120582119890 ge 119879 and (1 minus 120582)119890 lt 120591 Inthis case we have 119890 ge 119879

Strategy 3 Advertising effort assigned to LG is insufficientbut that to FG is sufficient that is 120582119890 lt 119879and (1 minus 120582)119890 ge 120591 Inthis case we have 119890 ge 120591

Strategy 4 Advertising effort assigned to both market seg-ments are insufficient that is 120582119890 lt 119879 and (1 minus 120582)119890 lt 120591 Inthis case we have 119890 lt 119879 + 120591

Here we consider the marketing strategy that the com-pany sells the product to both FG and LG in other words weassume 119863

119871(120596) gt 0 and 119863

119865(120596) gt 0 To facilitate presentation

we use 119895 = 1 2 3 4 to represent strategy 119895 All proofs of prop-ositions are relegated to the Appendix

In this paper we focus on deriving the local optimaladvertising and pricing policies as well as the respectiveassociated necessary conditions (and sufficient conditions aswell if it can be solved out analytically) for local optimality forindividual strategies (Due to the complexity of the problemit is very difficult to obtain the necessary and sufficient con-ditions for optimality for every strategyTherefore wemainlyfocus on exploring the necessary conditions for optimality)For any given market parameters the company can checkthe necessary conditions for each strategy If any one of thenecessary conditions of a particular strategy does not holdthen the local optimal of that particular strategy does notexist In other words the necessary conditions can be usedto screen out the strategies that never provide 120596lowast (the global


optimal solution) Afterwards the company can calculate thecorresponding companyrsquos profits amongst the strategies withlocal optimal policies satisfying the necessary conditions toidentify the global optimal strategy which is the one that givesthe maximum profit to the company

The total demand of the product is

119863 (120596) =

(119861 minus 119866 (119901 minus 119888) + 120572 (1 minus 119887) 119890 + 120582119873119868119890)

(1 + 119887120573)

(3)

and the associated companyrsquos profit is

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

(1 + 119887120573)

minus ℎ1198902minus 119898[119879 minus 120582119890]

+minus 120583[120591 minus (1 minus 120582) 119890]

+

(4)

where 119866 = (1 minus 119887)120574 + (1 + 120573)119892 119861 = 119883 minus 119866119888 and119873119868= 119886(1 +

120573) minus 120572(1 minus 119887)

31 Strategy 1 According to the basic conditions of Strategy1 the companyrsquos profit is

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573

minus ℎ1198902

(5)

We first study the optimal 120582 for Strategy 1

Proposition 2 For Strategy 1 (a) 120582lowast = 119879(119879+120591) if 119890 = 119879+120591(b) 120582lowast = 1 minus 120591119890 if 119873

119868gt 0 and 119890 gt 119879 + 120591 (c) 120582lowast = 119879119890 if

119873119868lt 0 and 119890 gt 119879+120591 (d) 120582lowast can take any value which satisfies

(1 minus 120582lowast)119890 ge 120591 and 120582lowast119890 ge 119879 if119873

119868= 0

Proposition 2 shows that the value of 120582lowast takes differentforms under various situations Specifically if 119890 = 119879 + 120591 then120582lowast= 119879(119879 + 120591) and hence 120582lowast119890 = 119879 and (1 minus 120582lowast)119890 = 120591 On

the one hand if the company wants to assign the minimumtotal advertising effort the optimal advertising effort assignedto individual market segments is just the minimum effortin individual market segments On the other hand if thecompany wants to assign a greater advertising effort (ie 119890 gt119879+120591) then the company should check the value of119873

119868first If

119873119868ge 0 then 120582lowast = 1 minus 120591119890 in turn 120582lowast119890 gt 119879 and (1 minus 120582lowast)119890 = 120591

Equivalently the advertising effort assigned to LG is higherthan the minimum requirement but the advertising effortassigned to FG is just sufficient for 119890 gt 119879 + 120591 and 119873

119868ge 0

If 119873119868le 0 then 120582

lowast119890 = 119879 and (1 minus 120582

lowast)119890 gt 120591 (Noting

that 120582lowast can take any value which satisfies (1 minus 120582lowast)119890 ge 120591

and 120582lowast119890 ge 119879 if 119873

119868= 0 However for simplicity we only

demonstrate here the two special cases 120582lowast satisfies (1minus120582lowast)119890 =120591 and 120582

lowast satisfies 120582lowast119890 = 119879 for 119873119868= 0) Therefore the

advertising effort assigned to FG is higher than theminimumrequirement but the advertising effort assigned to LG is justsufficient As a remark 119873

119868represents the sensitivity of 120582

to product demand Therefore a bigger 120582 advances productdemand as well as companyrsquos profit if119873

119868is positive a smaller

120582 advances product demand as well as companyrsquos profit if119873119868

is negative and the value of 120582 does not affect product demandand companyrsquos profit if119873

119868= 0

Following Proposition 2 we further consider three sub-strategies of Strategy 1

Strategy 1(a) 120582lowast = 119879(119879 + 120591) and 119890lowast = 119879 + 120591Strategy 1(b) 120582lowast = 1 minus 120591119890

lowast 119873119868ge 0 and 119890lowast gt 119879 + 120591

andStrategy 1(c) 120582lowast = 119879119890lowast119873

119868le 0 and 119890lowast gt 119879 + 120591

Notice that the above-mentioned conditions are specific forthe associated substrategies In particular they satisfy thefollowing three basic conditions of Strategy 1 120582119890 ge 119879 (1 minus120582)119890 ge 120591 and 119890 ge 119879 + 120591 We now need to check whether theyfulfill the remaining basic conditions of Strategy 1 namely119863119871(120596) gt 0 and119863

119865(120596) gt 0

Denoted by 120596lowast1198681(119894)

the local optimal advertising and pric-ing policy for Strategy 1(119894) where 119894 = 119886 119887 119888 Next we explorethe local optimal advertising and pricing policies for eachsubstrategy of Strategy 1

Strategy 1(a) Specific conditions 119863119871(120596) gt 0 119863

119865(120596) gt 0

120582lowast= 119879(119879 + 120591) and 119890lowast = 119879 + 120591By putting 119890lowast = 119879 + 120591 and 120582lowast = 119879(119879 + 120591) into (5) the

companyrsquos profit for Strategy 1(a) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

1 + 119887120573

+

(119901 minus 119888) [119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]

1 + 119887120573

minus ℎ(119879 + 120591)2

(6)

Notice that right hand side of (5) depends on 119901 only

Proposition 3 For Strategy 1(a) the local optimal advertis-ing and pricing policy exists only if (i) 119863

119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast

1198681(119886)) gt 0 (ii)119866 gt 0 and (iii)119861+119886(1+120573)119879+120572(1minus119887)120591 gt 0

Moreover if the local optimal advertising and pricing policy forStrategy 1(a) exists then it is unique and is given by

120596lowast

1198681(119886)= 119890lowast

1198681(119886)= 119879 + 120591 120582

lowast

1198681(119886)=

119879

(119879 + 120591)

119901lowast

1198681(119886)=

119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591

2119866

+ 119888

(7)

120587119871119871(120596lowast

1198681(119886)) =

[119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]2

4119866 (1 + 119887120573)

minus ℎ(119879 + 120591)2

(8)

The necessary conditions for the existence of 120596lowast1198681(119886)

areshown in Proposition 3 Specifically conditions in Propo-sition 3(i) are the basic conditions for any substrategy of


Strategy 1 The condition in Proposition 3(ii) ensures that thecompanyrsquos profit function is concave in 119901 whereas the one inProposition 3(iii) ensures that the constraint119901 gt 119888 is satisfiedMoreover Proposition 3 shows the explicit formulas of thelocal optimal advertising and pricing policy (if it exists) andthe associated companyrsquos profit for Strategy 1(a)

By considering the sensitivity of 119879 to 119890lowast1198681(119886)

120582lowast1198681(119886)

and119901lowast

1198681(119886) we find that 119890lowast

1198681(119886) 120582lowast1198681(119886)

and 119901lowast

1198681(119886)are all strictly

increasing in 119879 (because 119889119890lowast1198681(119886)

119889119879 = 1 gt 0 119889120582lowast1198681(119886)

119889119879 =

120591(119879 + 120591)2gt 0 and119889119901lowast

1198681(119886)119889119879 = 119886(1+120573)(2119866) gt 0) In other

words a bigger 119879 induces a higher optimal total advertisingeffort a bigger optimal proportion of advertising effortallocated to LG and a higher local optimal retail price of theproduct On the other hand 119890lowast

1198681(119886)is strictly increasing in 120591

and 120582lowast1198681(119886)

is strictly decreasing in 120591 Besides 119901lowast1198681(119886)

is strictlyincreasing in 120591 for 119887 lt 1 and 119901lowast

1198681(119886)is strictly decreasing in

120591 for 119887 gt 1 This shows that 119879 and 120591 affect 120596lowast1198681(119886)

differentlyInterestingly the values of 119898 and 120583 do not affect 120596lowast

1198681(119886)and

120587119871119871(120596lowast

1198681(119886)) An intuitive reason for this is that the advertising

effort assigned to both market segments is sufficient sothe penalties (119898 and 120583) for insufficient advertisings can beignored

Strategy 1(b) Specific conditions 119863119871(120596) gt 0 119863

119865(120596) gt 0

120582lowast= 1 minus 120591119890

lowast119873119868ge 0 and 119890lowast gt 119879 + 120591

By putting120582lowast = 1minus120591119890lowast into (5) the profit of the companyfor Strategy 1(b) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573

minus ℎ1198902

(9)

Proposition 4 For Strategy 1(b) the local optimal advertisingeffort (as a function of 119901) is given by

119890lowast

1198681(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) (10)

and 119890lowast1198681(119887)

(119901) is strictly increasing in 119901

Proposition 4 asserts that for Strategy 1(b) a higher retailprice induces a high optimal advertising effort Results ofProposition 4 are not surprising especially for luxury prod-ucts Advertising usually provides surpluses to luxury prod-ucts which are reflected by a higher retail price of the luxuryproduct

Proposition 5 Let 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)

2 Then forStrategy 1(b) the local optimal advertising and pricing policyexists only if (i) 119873

119868ge 0 119863

119871(120596lowast

1198681(119887)) gt 0 and 119863

119865(120596lowast

1198681(119887)) gt 0

(ii) 119884 gt 0 (iii) 119861 gt 119884(119879 + 120591)[119886(1 + 120573)] + 119873119868120591 Moreover if

the local optimal advertising and pricing policy for Strategy 1(b)exists then it is unique and is given by

120596lowast

1198681(119887)= 119890lowast

1198681(119887) 120582lowast

1198681(119887) 119901lowast

1198681(119887) (11)

120587119871119871(120596lowast

1198681(119887)) =

ℎ(119861 minus 119873119868120591)2

119884

(12)

where 119890lowast1198681(119887)

= 119886(1 + 120573)(119861 minus 119873119868120591)119884 120582lowast

1198681(119887)= 1 minus 120591119890

lowast

1198681(119887)

and 119901lowast119868119894(119887)

= 119888 + 2ℎ(119861 minus 119873119868120591)(1 + 119887120573)119884

Thenecessary conditions for120596lowast1198681(119887)

being finite are shownin Proposition 5 Specifically conditions in Proposition 5(i)are the specific conditions of Strategy 1(b) The conditionin Proposition 5(ii) ensures that the profit function of thecompany for Strategy 1(b) is concave in 119901 The condition inProposition 5(iii) ensures 119890 gt 119879 + 120591 which is another basiccondition for Strategy 1(b) Notice that 119901lowast

1198681(119887)gt 119888 if and only

if 119861 gt 119873119868120591 Because 119861 gt 119884(119879 + 120591)[119886(1 + 120573)] + 119873

119868120591 implies

that 119861 gt 119873119868120591 the condition in Proposition 5(iii) does cover

the constraint 119901 gt 119888 Moreover Proposition 5 provides theexplicit formulas of the local optimal advertising and pricingpolicy (if it exists) and the associated companyrsquos profit forStrategy 1(b)

According to Proposition 5 119890lowast1198681(119886)

120582lowast1198681(119886)

119901lowast1198681(119886)

and120587119871119871(120596lowast

1198681(119887)) are decreasing in 120591(119889120582

lowast

1198681(119886)119889119879 = minus119884119861119886(1 +

120573)(119861 minus 119873119868120591)2lt 0 and the other derivatives are trivial but

independent of 119879 In other words for Strategy 1(b) a bigger 120591induces a lower advertising effort a smaller portion of adver-tising effort allocated to LG and a lower optimal retail priceof the product However 119879 does not affect 119890lowast

1198681(119886) 120582lowast1198681(119886)

119901lowast

1198681(119886) and 120587

119871119871(120596lowast

1198681(119887))

Finally similar to Strategy 1(a) the value of 119898 and 120583 donot affect 120596lowast

1198681(119887)and 120587

119871119871(120596lowast

1198681(119887)) as the advertising budgets

assigned to both market segments are sufficient and hencethe penalties 119898 and 120583 for insufficient advertisings can beignored for Strategy 1(b)

Strategy 1(c) Specific conditions 119863119871(120596) gt 0 119863

119865(120596) gt 0

119873119868le 0 119890lowast gt 119879 + 120591 and 120582lowast = 119879119890lowastFirst of all 119873

119868le 0 is equivalent to 119886(1 + 120573) le 120572(1 minus

119887) which implies 119887 lt 1 as 119886(1 + 120573) gt 0 Then by putting120582lowast= 119879119890

lowast into (5) the profit of the company for Strategy 1(c)becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573

minus ℎ1198902

(13)

Similar to Strategy 1(b) we obtain the following results forStrategy 1(c)


Proposition 6 Let 119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)

2 Then forStrategy 1(c) (a) the local optimal advertising effort (as a func-tion of 119901) is given by

119890lowast

1198681(119888)(119901) =

120572 (1 minus 119887)

2ℎ (1 + 119887120573)

(119901 minus 119888) (14)

and 119890lowast1198681(119888)

(119901) is strictly increasing in 119901(b)The local optimal advertising and pricing strategy exists

only if (i) 119873119868le 0 119863

119871(120596lowast

1198681(119888)) gt 0 and 119863

119865(120596lowast

1198681(119888)) gt 0 (ii)

119885 gt 0 (iii) 119861 gt 119885(119879 + 120591)[120572(1 minus 119887)] minus 119873119868119879 Moreover if

the local optimal advertising and pricing policy for Strategy 1(c)exists then it is unique and is given by

120596lowast

1198681(119888)= 119890lowast

1198681(119888) 120582lowast

1198681(119888) 119901lowast

1198681(119888) (15)

120587119871119871(120596lowast

1198681(119888)) =

ℎ(119861 + 119873119868119879)2

119885

(16)

where 119890lowast1198681(119888)

= 120572(1minus119887)(119861+119873119868119879)119885 120582lowast

1198681(119888)= 119879119885120572(1minus119887)(119861+

119873119868119879) and 119901lowast

1198681(119888)= 2ℎ(119861 + 119873

119868119879)(1 + 119887120573)119885 + 119888

Similar to Propositions 4 and 5 for Strategy 1(b) Proposi-tion 6(a) shows that the optimal advertising effort is increas-ing in the optimal retail price for Strategy 1(c) and the neces-sary conditions for 120596lowast

1198681(119888)being finite are shown in Proposi-

tion 6(b) Specifically conditions in Proposition 6(b)(i) arethe specific conditions of Strategy 1(c) The condition inProposition 6(b)(ii) ensures that the profit function of thecompany for Strategy 1(c) is concave in 119901 whereas the onein Proposition 6(b)(iii) ensures 119890 gt 119879 + 120591 which is anotherbasic condition for Strategy 1(c) Notice that 119901lowast

119868119894(119888)gt 119888 if

119861 gt 119885(119879 + 120591)[120572(1 minus 119887)] minus 119873119868119879 Moreover Proposition 6(b)

provides the explicit formulas of the local optimal advertisingand pricing policy (if it exists) and the associated companyrsquosprofit for Strategy 1(c)

According to Proposition 6 120587119871119871(120596lowast

1198681(119888)) 119890lowast1198681(119888)

and 119901lowast1198681(119888)

are decreasing in119879 while 120582lowast1198681(119888)

is increasing in119879This showsthat the company will spend less on advertising will be morefocused on LG and will set a lower retail price of the productwhen119879 increases Although there is less expenditure spent onadvertising the company still loses profit due to a lower retailprice of the product offered to the customers On the otherhand 119890lowast

1198681(119888) 120582lowast1198681(119888)

and 119901lowast1198681(119888)

are independent of 120591 Similarlythese findings are caused by the technical issue as 119879 does notinclude in any specific condition of Strategy 1(c)

Finally similar to Strategies 1(a) and 1(b) the values of119898and 120583 do not affect 120596lowast

1198681(119888)and 120587

119871119871(120596lowast

1198681(119888)) as the advertising

effort assigned to both market segments are sufficient andhence the penalties 119898 and 120583 for insufficient advertising canbe ignored for Strategy 1(c)


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573

minus ℎ1198902minus 120583 [120591 minus (1 minus 120582) 119890]

(17)

Proposition 7 For Strategy 2 (a) the local optimal advertisingand pricing policy does not satisfy119873

119868(119901 minus 119888) lt 120583(1 + 119887120573) and

119890 ge 119879 + 120591 simultaneously (b) 120582lowast = 1 if 119890 = 119879 (c) 120582lowast = 1 if119873119868(119901 minus 119888) gt 120583(1 + 119887120573) and 119890 gt 119879 (d) 120582lowast = 119879119890 if119873

119868(119901 minus 119888) lt

120583(1 + 119887120573) and 119879 lt 119890 lt 119879 + 120591 and (e) multiple 120582lowast exist if119873119868(119901 minus 119888) = 120583(1 + 119887120573) and 119890 gt 119879

Proposition 7 asserts that we can ignore all the cases forwhich 119873

119868(119901 minus 119888) lt 120583(1 + 119887120573) and 119890 ge 119879 + 120591 under Strategy

2 It also shows that the value of 120582lowast varies under differentsituations Specifically if the company wants to assign theadvertising effort such that 119890 = 119879 we have 120582lowast = 1 Howeverif the companywants to assign the advertising effort such that119890 gt 119879 then the company should take into account the valuesof119873119868(119901minus119888) and120583(1+119887120573) In particular if119873

119868(119901minus119888) gt 120583(1+119887120573)

then 120582lowast = 1 if 119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120582lowast = 119879119890 if

119873119868(119901 minus 119888) = 120583(1 + 119887120573) then for any given 119890 gt 119879 any 120582 that

satisfies (1 minus 120582)119890 lt 120591 and 120582119890 ge 119879 is optimal for Strategy 2In other words if 119873

119868(119901 minus 119888) gt 120583(1 + 119887120573) then the company

should assign its advertising effort to LG only If119873119868(119901 minus 119888) lt

120583(1+119887120573) then the company should assign to LG itsminimumrequired advertising effort 119879 whereas to FG it should assignthe advertising effort less than 120591 theminimumrequired effortfor the group Lastly if 119873

119868(119901 minus 119888) = 120583(1 + 119887120573) then the

company can select any120582 that satisfies (1minus120582)119890 lt 120591 and120582119890 ge 119879As 120582 = 1 satisfies (1 minus 120582)119890 lt 120591 and 120582119890 ge 119879 for any given gt 119879we take 120582lowast = 1 for 119873

119868(119901 minus 119888) = 120583(1 + 119887120573) According to the

previous discussions we further consider four substrategiesof Strategy 2

Strategy 2(a) 119890 = 119879 120582lowast = 1

Strategy 2(b)119873119868(119901minus119888) gt 120583(1+119887120573) 119890 gt 119879 and 120582lowast = 1

Strategy 2(c)119873119868(119901minus119888) = 120583(1+119887120573) 119890 gt 119879 and 120582lowast = 1

and

Strategy 2(d)119873119868(119901 minus 119888) lt 120583(1 + 119887120573) 119879 lt 119890 lt 119879 + 120591

and 120582lowast = 119879119890

The above-mentioned conditions are specific for the asso-ciated substrategies In particular they satisfy the followingthree basic conditions of Strategy 2 120582119890 ge 119879 (1 minus 120582)119890 lt 120591 and119890 ge 119879 Next we need to check whether they also fulfill theremaining basic conditions of Strategy 2 namely 119863

119871(120596) gt 0

and119863119865(120596) gt 0

Denote by 120596lowast1198682(119894)

for 119894 = 119886 119887 119888 119889 the local optimal adver-tising and price policy for Strategy 2(119894) Next we explore thelocal optimal advertising and pricing policies for individualsubstrategies


119865(120596) gt 0

119890 = 119879 and 120582lowast = 1By putting 119890 = 119879 and 120582lowast = 1 into (17) the profit of the

company for Strategy 2(a) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119879]

1 + 119887120573

minus ℎ1198792minus 120583120591

(18)


Proposition 8 For Strategy 2(a) the local optimal advertisingand pricing policy exists only if (i) 119863

119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 + 119886(1 + 120573)119879 gt 0


120596lowast

1198682(119886)= 119890lowast

1198682(119886)= 119879 120582

lowast

1198682(119886)= 1

119901lowast

1198682(119886)= 119888 +

[119861 + 119886 (1 + 120573) 119879]

(2119866)

(19)

120587119871119871(120596lowast

1198682(119886)) =

1198612+ 2119861119886 (1 + 120573) 119879 minus 119884119879

2

4119866 (1 + 119887120573)

minus 120583120591 (20)

The necessary conditions for120596lowast1198682(119886)

being finite are shownin Proposition 8 Specifically conditions in Proposition 8(i)are the basic conditions of Strategy 2The condition in Propo-sition 8(ii) ensures that the profit function of the companyfor Strategy 2(a) is concave in 119901 whereas the conditionin Proposition 8(iii) ensures that 119901lowast

1198682(119886)gt 119888 Moreover

Proposition 8 provides the explicit formulas of the localoptimal advertising and pricing strategy and the associatedcompanyrsquos profit for Strategy 2(a) Furthermore 119890lowast

1198682(119886)and

119901lowast

1198682(119886)are strictly increasing in 119879 In other words for Strategy

2(a) a bigger 119879 induces a higher optimal advertising effortand a higher retail price of the product On the other handaccording to Proposition 8 we know that 119890lowast

1198682(119886)and 119901lowast

1198682(119886)are

independent of 120591


119865(120596) gt 0

119873119868(119901 minus 119888) gt 120583(1 + 119887120573) 119890 gt 119879 and 120582lowast = 1By putting 120582lowast = 1 into (17) the profit of the company for

Strategy 2(b) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 120583120591

(21)

As 119901 gt 119888 and 120583(1 + 119887120573) gt 0 119873119868(119901 minus 119888) gt 120583(1 + 119887120573) implies

119873119868gt 0

Proposition 9 For Strategy 2(b) (a) the local optimal adver-tising effort is given by

119890lowast

1198682(119887)(119901) = 119890

lowast

1198681(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(22)

and 119890lowast1198682(119887)

(119901) is strictly increasing in 119901(b) The local optimal advertising and pricing policy exists

only if (i) 119873119868gt 0 119863

119871(120596lowast

1198682(119887)) gt 0 and 119863

119865(120596lowast

1198682(119887)) gt 0 (ii)

119884 gt 0 (iii) 119861 gt 119879119884[119886(1 + 120573)] and (iv) 119861 ge 120583119884(2ℎ119873119868)

Moreover if the local optimal advertising and pricing policy forStrategy 2(b) exists then it is unique and is given by

120596lowast

1198682(119887)= 119890lowast

1198682(119887) 120582lowast

1198682(119887) 119901lowast

1198682(119887) (23)

120587119871119871(120596lowast

1198682(119887)) =

ℎ1198612

119884

minus 120583120591 (24)

where 119890lowast1198682(119887)

= 119886(1 + 120573)119861119884 120582lowast1198682(119887)

= 1 and 119901lowast1198682(119887)

= 2ℎ119861(1 +

119887120573)119884 + 119888

Proposition 9(a) shows that for Strategy 2(b) high retailprice should be supported by high advertising effort of thecompany

The necessary conditions for 120596lowast

1198682(119887)being finite are

shown in Proposition 9(b) Specifically conditions in Propo-sition 9(b)(i) are the specific conditions of Strategy 2(b) Thecondition in Proposition 9(b)(ii) ensures that the profit func-tion of the company for Strategy 2(b) is concave in 119901 whereasthat in Proposition 9(b)(iii) ensures 119890 gt 119879 which is the basiccondition for Strategy 2(b) The condition in Proposi-tion 9(b)(iv) ensures that 119873

119868(119901 minus 119888) gt 120583(1 + 119887120573) which is

another basic condition for Strategy 2(b) noting that theconstraint 119901lowast

1198682(119887)gt 119888 is fulfilled under the condition in

Proposition 9(b)(iii) Moreover Proposition 9(b) providesthe explicit formulas of the local optimal advertising andpricing policy and the associated companyrsquos profit for Strategy2(b) Furthermore 120587

119871119871(120596lowast

1198682(119887)) is strictly decreasing in 120591 and

120583 but is independent of 119879 and119898Interestingly as shown by Proposition 9(b) 119890lowast

1198682(119887) 120582lowast1198682(119887)

and 119901

lowast

1198682(119887)are independent of 120591 and 119879 These findings are

different from previous findings in which both the optimaladvertising effort and the optimal retail price depend oneither 120591 or 119879 On the other hand 119890lowast

1198682(119887) 120582lowast1198682(119887)

and 119901lowast1198682(119887)

areindependent of 120583 and119898


119865(120596) gt 0

119873119868(119901 minus 119888) = 120583(1 + 119887120573) 119890 gt 119879 and 120582lowast = 1As 119901 gt 119888 and 120583(1+119887120573) gt 0119873

119868(119901minus 119888) = 120583(1+119887120573) implies

119873119868gt 0

Proposition 10 For Strategy 2(c) (a) the local optimal adver-tising effort as a function of 119901 is given by

119890lowast

1198682(119888)(119901) = 119890

lowast

1198682(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(25)

and 119890lowast1198682(119888)


only if (i)119873119868gt 0119863

119871(120596lowast

1198682(119888)) gt 0 and119863

119865(120596lowast

1198682(119888)) gt 0 and (ii)

119886120583(1 + 120573) gt 2ℎ119873119868119879 Moreover if the local optimal advertising

and pricing policy for Strategy 2(c) exists then it is unique andis given by

120596lowast

1198682(119888)= 119890lowast

1198682(119888) 120582lowast

1198682(119888) 119901lowast

1198682(119888) (26)

120587119871119871(120596lowast

1198682(119888)) =

119861120583

119873119868

minus

1198841205832

4ℎ1198732

119868

minus 120583120591 (27)

where 119890lowast1198682(119888)

= 119886120583(1 + 120573)(2ℎ119873119868) 120582lowast1198682(119888)

= 1 and 119901lowast1198682(119888)

= 119888 +

120583(1 + 119887120573)119873119868


Proposition 10(a) asserts that for Strategy 2(c) a highretail price should be supported by a high advertising effortof the brand


being finite are shownin Proposition 10(b) Specifically conditions in Proposi-tion 10(b)(i) are the specific conditions of Strategy 2(c)Proposition 10(b) also provides the explicit formulas of thelocal optimal advertising and pricing policy and the asso-ciated companyrsquos profit for Strategy 2(c) Furthermore120587119871119871(120596lowast

1198682(119888)) is strictly decreasing in 120591 and independent of 119879

and119898As shown in Proposition 10(b) 119890lowast

1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)

are independent of 120591 and 119879 These findings are similar tothose for Strategy 2(b) On the other hand 119890lowast

1198682(119888)and119901lowast

1198682(119888)are

increasing in 120583 but are independent of119898 Moreover 120582lowast1198682(119888)

isindependent of 120583 and119898

Strategy 2(d) Specific conditions 119863119871(120596) gt 0 119863

119865(120596) gt 0

119873119868(119901 minus 119888) lt 120583(1 + 119887120573) 119879 lt 119890 lt 119879 + 120591 and 120582lowast = 119879119890By putting 120582lowast = 119879119890 into (17) the profit of the company

for Strategy 2(d) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573

minus ℎ1198902minus 120583 [119879 + 120591 minus 119890]

(28)

Proposition 11 For Strategy 2(d) (a) the local optimal adver-tising effort as a function of retail price 119901 is given by

119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)

(b) The local optimal advertising and pricing policy exists onlyif (i) 119863

119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887)] le 120583119885 Moreover if the local

optimal advertising and pricing policy for Strategy 2(d) existsthen it is unique and is given by

120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]

Noting that 119890lowast1198682(119889)

(119901) is strictly increasing in 119901 only if 119887 lt1 If 119887 gt 1 then 119890lowast

1198682(119889)(119901) is strictly decreasing in 119901Therefore

Proposition 11(a) asserts that a high advertising effort of thecompany does not always support a high retail price and itdoes happen if 119887 gt 1 for Strategy 2(d)


being finite are shownin Proposition 11(b) Specifically conditions in Proposi-tion 11(b)(i) are the basic conditions of Strategy 2 whilst thatin Proposition 11(b)(ii) ensures that the profit function of thecompany for Strategy 2(d) is concave in 119901 The condition inProposition 11(b)(iii) ensures 119879 lt 119890 lt 119879 + 120591 which is oneof the specific conditions of Strategy 2(d) whereas the onein Proposition 11(b)(iv) ensures that 119901lowast

1198682(119889)gt 119888 Finally the

condition in Proposition 11(b)(v) ensures that 119873119868(119901 minus 119888) lt

120583(1+119887120573)which is another specific condition of Strategy 2(d)Moreover Proposition 11(b) provides the explicit formulasof the local optimal advertising and pricing policy and theassociated companyrsquos profit for Strategy 2(d)

By considering the sensitivities of 119890lowast1198682(119889)

and 119901lowast

1198682(119889) we

observe that 119890lowast1198682(119889)

and 119901lowast1198682(119889)

are increasing in 119879 if 119873119868gt 0

decreasing in 119879 if119873119868lt 0 whilst they are independent of 119879 if

119873119868= 0 In other words a bigger 119879 induces a bigger optimal

advertising effort and a higher optimal retail price for Strategy2(d) if119873

119868is positive By contrast a bigger119879 induces a smaller

optimal advertising effort and a smaller optimal retail pricefor Strategy 2(d) if119873

119868is negative Regarding the sensitivities

of 120591 to 119890lowast

1198682(119889) 120582lowast1198682(119889)

and 119901lowast

1198682(119889) from Proposition 15 it is

obvious that 119890lowast1198682(119889)

120582lowast1198682(119889)

and 119901lowast1198682(119889)

are independent of 120591

33 Strategy 3 According to the basic conditions of Strategy3 the companyrsquos profit for Strategy 3 is

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573

minus ℎ1198902minus 119898 (119879 minus 120582119890)

(32)

We first study the optimal solution of 120582 for Strategy 3

Proposition 12 For Strategy 3 (a) the local optimal advertis-ing and pricing policy does not satisfy119873

119868(119901 minus 119888) gt minus119898(1 + 119887120573)

and 119890 ge 119879 + 120591 simultaneously (b) if 119890 = 120591 then 120582lowast = 0 (c) if119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) and 120591 lt 119890 lt 119879+ 120591 then 120582lowast = 1 minus 120591119890

(d) if 119873119868(119901 minus 119888) lt minus119898(1 + 119887120573) and 119890 gt 120591 then 120582lowast = 0 (e)

multiple 120582lowast exist if119873119868(119901 minus 119888) = minus119898(1 + 119887120573) and 119890 gt 119879

Proposition 12 asserts that we can ignore all the cases forwhich119873

119868(119901 minus 119888) gt minus119898(1 + 119887120573) and 119890 ge 119879 + 120591 under Strategy

3 Moreover Proposition 12 shows that the value of 120582lowast variesunder different situations Specifically if the company wantsto exert its advertising effort as 119890 = 120591 then the companyshould assign advertising effort to FG only that is 120582lowast = 0However if the company wants to exert an advertising effortgreater than 120591 (ie 119890 gt 120591) then the company should checkthe value of119873

119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)

then 120582lowast = (1minus120582)119890 If119873119868(119901minus119888) lt minus119898(1+119887120573) then 120582lowast = 0 If

119873119868(119901minus119888) = minus119898(1+119887120573) then there exist multiple 120582lowast In other

words if119873119868(119901 minus 119888) lt minus119898(1 + 119887120573) then the company should

assign the advertising effort to FG only However if 119873119868(119901 minus

119888) gt minus119898(1 + 119887120573) then the company should assign to FGthe minimum advertising resource for the group (ie 120591) andto LG it should assign the advertising effort less than theminimum advertising resource for the group (ie 119879)


As119901 gt 119888 and119898(1+119887120573) gt 0119873119868(119901minus119888) le minus119898(1+119887120573) implies

that 119873119868lt 0 and hence 119887 lt 1 Furthermore for 119873

119868(119901 minus 119888) =

120583(1 + 119887120573) the company can select any 120582 that satisfies (1 minus120582)119890 lt 120591 and 120582119890 ge 119879 As 120582 = 0 satisfies (1 minus 120582)119890 ge 120591 and120582119890 lt 119879 for any given 119890 gt 119879 we take 120582lowast = 0 for 119873

119868(119901 minus 119888) =

minus119898(1 + 119887120573) Further to the previous discussions we considerfour substrategies of Strategy 3

Strategy 3(a) 119890 = 120591 and 120582lowast = 0Strategy 3(b)119873

119868(119901 minus 119888) gt minus119898(1 + 119887120573) 120591 lt 119890 lt 119879 + 120591

and 120582lowast = 1 minus 120591119890Strategy 3(c) 119887 lt 1119873

119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0Strategy 3(d) 119887 lt 1119873

119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0

The above-mentioned conditions are specific to the associ-ated substrategies In particular they satisfy the three basicconditions of Strategy 3 120582119890 lt 119879 (1minus120582)119890 ge 120591 and 119890 ge 120591 Nextweneed to considerwhether these substrategies also fulfill theremaining basic conditions of Strategy 3 namely 119863

119871(120596) gt 0

and119863119865(120596) gt 0

We denote by 120596lowast1198683(119894)

for 119894 = 119886 119887 119888 119889 the local optimaladvertising and pricing policy for Strategy 3(119894) We nowproceed to explore the local optimal advertising and pricingpolicies for individual substrategies

Strategy 3(a) Specific conditions119863119871(120596) gt 0 and119863

119865(120596) gt 0



120587119871119871(120596) =

(119901 minus 119888) [119861 + 120572 (1 minus 120573) 120591] minus 119866(119901 minus 119888)2

1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast

1198683(119886)) gt 0 (ii)119866 gt 0 and (iii)119861+120572(1minus119887)120591 gt 0Moreover

if the local optimal advertising and pricing policy for Strategy3(a) exists then it is unique and is given by

120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)


being finite are shownin Proposition 13 Specifically conditions in Proposition 13(i)are the basic conditions of Strategy 3 and the conditionin Proposition 13(ii) ensures that the profit function ofthe company for Strategy 3(a) is concave in 119901 whilst thatin Proposition 13(iii) ensures that 119901lowast

1198683(119886)gt 119888 Moreover

Proposition 13 shows the explicit formulas of the local optimal

advertising and pricing policy as well as the associatedcompanyrsquos profit for Strategy 3(a)

For the sensitivity of 120591 to 119890lowast1198683(119886)

and 119901lowast1198683(119886)

from Propo-sition 13 we find that 119890lowast

1198683(119886)is always strictly increasing in 120591

119901lowast

1198683(119886)is strictly increasing in 120591 only if 119887 lt 1 if 119887 gt 1 then

119901lowast

1198683(119886)is strictly decreasing in 120591 In other words for Strategy

3(a) a bigger 120591 always induces a higher optimal advertisingeffort A bigger 120591 also induces a higher retail price of theproduct if 119887 gt 1 but it induces a lower retail price of theproduct if 119887 lt 1


119865(120596) gt 0

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 120591 lt 119890 lt 119879 + 120591 and 120582lowast = 1 minus 120591119890By putting120582lowast = 1minus120591119890 into (32) the profit of the company

for Strategy 3(b) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573

minus ℎ1198902minus 119898 (119879 minus 119890 + 120591)

(36)

Proposition 14 For Strategy 3(b) (a) the local optimaladvertising effort as a function of retail price 119901 is given by

119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)

119861 gt 119873119868120591minus119886119898(1+120573)(2ℎ) (iv) (Θ120591minus2mG(1+119887120573))119886(1+120573) lt

119861 lt (119884119879+Θ120591minus2119898119866(1+119887120573))119886(1+120573) and (v) [2ℎ(119861minus119873119868120591)+

119886119898(1+120573)]119873119868gt minus119898119884 whereΘ = 4ℎ119866(1+119887120573)minus119886120572(1+120573)(1minus119887)

and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)

2 Moreover (c) if the localoptimal advertising and pricing policy for Strategy 3(b) existsthen it is unique and is given by

120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884

Proposition 14(a) implies that for Strategy 3(b) a highoptimal retail price results in a high optimal advertisingeffort Proposition 14(c) shows the explicit formulas of thelocal optimal advertising and pricing policy and the associ-ated companyrsquos profit for Strategy 3(b) Moreover conditionsin Proposition 14(b)(i) are the basic conditions of Strategy3 the condition in Proposition 14(b)(ii) ensures that theprofit function of the company for Strategy 3(b) is concavein 119901 whilst that in Proposition 14(b)(iii) ensures 119901 gt 119888 the


condition in Proposition 14(b)(iv) ensures that 120591 lt 119890 lt 119879 + 120591which is the specific condition of Strategy 3(b) Finally thecondition in Proposition 14(b) ensures that 119873

119868(119901 minus 119888) gt

minus119898(1 + 119887120573) which is also the specific condition for Strategy3(b)

Considering the sensitivities of 119879 with respect to 119890lowast1198683(119887)

120582lowast

1198683(119887) and 119901

lowast

1198683(119887) we observe that all 119890lowast

1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873

119868 A bigger 119879 induces a smaller optimal

advertising effort and a lower optimal retail price for Strategy3(b) if 119873

119868is positive On the contrary a bigger 119879 induces a

bigger optimal advertising effort and a higher optimal retailprice for Strategy 3(b) if 119873

119868is negative Finally a bigger 119879

induces a smaller 120582lowast1198683(119887)

if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573) 119890 gt 120591 and 120582lowast = 0

By putting 120582lowast = 0 into (32) the profit of the company forStrategy 3(c) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)

Proposition 15 For Strategy 3(c) (a) the local optimal adver-tising effort as a function of retail price 119901 is given by

119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt

0 (iii) 119861 gt 120591119885[120572(1 minus 119887)] and (iv) 119861 gt minus119898119885(2ℎ119873119868) where

119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)

2 Moreover if the local optimaladvertising and pricing policy for Strategy 3(c) exists then it isunique and is given by

120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)

Proposition 15(a) shows that for Strategy 3(c) a higherretail price induces a higher optimal advertising effort ofthe brand Proposition 15(b) shows the explicit formulas ofthe local optimal advertising and pricing policy and theassociated companyrsquos profit for Strategy 3(c)Moreover it alsostates the necessary conditions for having a finite 120596lowast

1198683(119888) Plus

according to Proposition 15(b) 119890lowast1198683(119888)

120582lowast1198683(119888)

and 119901lowast1198683(119888)

are allindependent of 119879 120591119898 and 120583

Strategy 3(d) Specific conditions119863119871(120596) gt 0 and119863

119865(120596) gt 0

119887 lt 1119873119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573) 119890 gt 120591 and 120582lowast = 0

Proposition 16 For Strategy 3(d) (a) the local optimal adver-tising effort as a function of 119901 is given by

119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)

(119901) is strictly increasing in 119901(b) the local optimal advertising and pricing policy exists

only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)

120572119898(119887 minus 1) lt 2ℎ119873119868120591 Moreover if the local optimal advertising

and pricing policy for Strategy 3(d) exists then it is unique andis given by

120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)

Proposition 16(a) asserts that for Strategy 3(d) a highretail price induces a high optimal advertising effort of thebrand Proposition 16(b) provides the explicit formulas of thelocal optimal advertising and pricing policy and the associ-ated companyrsquos profit for Strategy 3(d) Moreover the neces-sary conditions for having a finite 120596lowast

1198683(119889)are shown in Propo-

sition 16(b) Specifically conditions in Proposition 16(b)(i)are the specific conditions of Strategy 3(d) whereas thecondition in Proposition 16(b)(ii) ensures 119890 gt 120591 which isalso the specific condition for Strategy 3(d) As shown inProposition 16(b) 119890lowast

1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)

are independentof 120591 119879 and 120583 On the other hand 119890lowast

1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898

34 Strategy 4 Basic conditions119863119871(120596) gt 0119863

119865(120596) gt 0 120582119890 lt

119879 (1 minus 120582)119890 lt 120591 and 119890 lt 119879 + 120591The companyrsquos profit for Strategy 4 is

120587119871119871(120596) =

(119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890] minus 119866(119901 minus 119888)

2

1 + 119887120573

minus ℎ1198902minus 119898119879 minus 120583120591 + (119898120582 + 120583 minus 120583120582) 119890

(47)

Proposition 17 (a) When119873119868(119901 minus 119888) gt (120583 minus 119898)(1 + 119887120573) and

119879 le 119890 lt 119879 + 120591 Strategy 2 dominates Strategy 4 (b) When119873119868(119901 minus 119888) lt (120583 minus 119898)(1 + 119887120573) and 120591 le 119890 lt 119879 + 120591 Strategy 3

dominates Strategy 4 (c)When119873119868(119901minus119888) = (120583minus119898)(1+119887120573) at

least one of Strategies 2 and 3 dominates Strategy 4

Proposition 17 states the three cases under which Strategy4 will never be the global optimal advertising and pricingstrategy Next we focus on the remaining cases


Proposition 18 For Strategy 4 (a) if119873119868(119901 minus 119888) gt (120583 minus119898)(1 +

119887120573) and 0 lt 119890 lt 119879 then 120582lowast = 1 (b) If119873119868(119901minus 119888) lt (120583minus119898)(1+

119887120573) and 0 lt 119890 lt 120591 then 120582lowast = 0

Proposition 17 shows that the value of 120582lowast differs underdifferent situations Note that the value of 120582 can be ignored if119890 = 0 If the company wants to assign advertising effort (ie119890 gt 0) then the company should check the value of 119890 and119873119868(119901 minus 119888) + 119898(1 + 119887120573) If 119873

119868(119901 minus 119888) gt (120583 minus 119898)(1 + 119887120573) and

0 lt 119890 lt 119879 then the company should assign its advertisingeffort to LG only If119873

119868(119901minus119888) lt (120583minus119898)(1+119887120573) and 0 lt 119890 lt 120591

then the company should assign its advertising effort to FGonly

We further consider the following three substrategies ofStrategy 4

Strategy 4(a) 119890lowast = 0

Strategy 4(b)119873119868(119901 minus 119888) gt (120583 minus 119898)(1 + 119887120573) 0 lt 119890 lt 119879

and 120582lowast = 1

Strategy 4(c)119873119868(119901 minus 119888) lt (120583 minus 119898)(1 + 119887120573) 0 lt 119890 lt 120591


The above-mentioned conditions are specific for the asso-ciated substrategies In particular they satisfy the followingthree basic conditions of Strategy 4 120582119890 lt 119879 (1 minus 120582)119890 lt 120591and 0 le 119890 lt 119879 + 120591 We proceed to check whether they alsofulfill the remaining basic conditions of Strategy 4 namely119863119871(120596) gt 0 and119863

119865(120596) gt 0

Denote by 120596lowast

1198684(119894) for 119894 = 119886 119887 119888 the local optimal

advertising and price policy for Strategy 4(119894) In the followingwe explore the local optimal advertising and pricing policiesfor individual substrategies

Strategy 4(a) Specific conditions 119890lowast = 0By putting 119890lowast = 0 into (47) the profit of the company for

Strategy 4(a) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573

minus ℎ1198902minus 119898119879 minus 120583120591 (48)


119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast

1198684(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 gt 0 Moreover if the

local optimal advertising and pricing policy exists then it isunique and is given by

120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)

Similar to the other strategies the necessary conditionsfor having a finite 120596lowast

1198684(119886)are shown in Proposition 19

Strategy 4(b) Specific conditions119873119868(119901minus119888) gt (120583minus119898)(1+119887120573)

0 lt 119890 lt 119879 and 120582lowast = 1

By putting 120582lowast = 1 into (47) the profit of the company forStrategy 4(b) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879 minus 120583120591 + 119898119890

(51)

Proposition 20 For Strategy 4(b) (a) the local optimal adver-tising effort as a function of retail price 119901 is given by

119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)

119861 gt minus119886119898(1+120573)(2ℎ) (condition for119901 gt 119888) (iv)minus2119898119866(1+119887120573)119886(1+120573) lt 119861 lt (119879119884minus2119898119866(1+119887120573))119886(1+120573) and (v) 2ℎ119873

119868119861 gt

120583119884minus119898Θ Moreover if the local optimal advertising and pricingpolicy for Strategy 4(b) exists then it is unique and is given by

120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888

Proposition 20(a) shows that for Strategy 4(b) a higherretail price induces a higher advertising effort from thecompany The necessary conditions for having a finite 120596lowast

1198684(119887)

are shown in Proposition 20(b) In particular conditions inProposition 20(b)(i) are the basic conditions for Strategy 4(b)the condition in Proposition 20(b)(ii) ensures that the profitfunction of the company for Strategy 4(b) is concave in 119901the one in Proposition 20(b)(iii) ensures 119901 gt 119888 conditionsin Proposition 20(b)(iv) and (v) ensure that 0 lt 119890 lt 119879 and119873119868(119901minus119888) gt (120583minus119898)(1+119887120573) respectively which are the specific

conditions for Strategy 4(b) Proposition 20(b) also shows theexplicit formulas of the local optimal advertising and pricingpolicy and the associated companyrsquos profit for Strategy 4(b)Note that 119890lowast

1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)

are independent of 119879 and120591

Strategy 4(c) Specific conditions119873119868(119901minus119888) lt (120583minus119898)(1+119887120573)

0 lt 119890 lt 120591 and 120582lowast = 0By putting 120582lowast = 0 into (47) the profit of the company for

Strategy 4(c) becomes

120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)


Proposition 21 For Strategy 4(c) the local optimal advertis-ing effort as a function of retail price 119901 is given by

119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)

(119901) is strictly increasing in 119901 only if 119887 lt 1if 119887 gt 1 then 119890lowast

1198684(119888)(119901) is strictly decreasing in 119901 Therefore

Proposition 21 asserts that a higher retail pricemaynot inducea greater advertising effort unless when 119887 lt 1 for Strategy4(c)

Proposition 22 For Strategy 4(c) the local optimal advertis-ing and pricing policy exists only if (i) 119863

119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast

1198684(119888)) gt 0 (ii) 119885 gt 0 (iii) 2ℎ119861 + 120572120583(1 minus 119887) gt 0 (iv)

0 lt 2120583119866(1+119887120573)+120572(1minus119887)119861 lt 119885120591 and (v) 2ℎ119861119873119868lt Θ120583minus119885119898

Moreover if the local optimal advertising and pricing policy forStrategy 4(c) exists then it is unique and is given by

120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888

The necessary conditions for having a finite 120596lowast1198684(119888)

areshown in Proposition 22 Specifically conditions in Propo-sition 22(i) are the basic conditions for Strategy 4 the onein Proposition 22(ii) ensures that the profit function ofthe company for Strategy 4(c) is concave in 119901 the one inProposition 22(iii) ensures that 119901 gt 119888 finally the conditionsin Proposition 22(iv) and (v) ensure that 0 lt 119890 lt 120591 and119873

119868(119901minus

119888) lt (120583 minus 119898)(1 + 119887120573) which are the specific conditions forStrategy 4(c) Moreover Proposition 22 shows the explicitformulas of the local optimal advertising and pricing policyand the associated companyrsquos profit for Strategy 4(c) Noticethat 119890lowast1198684(119888)

120582lowast1198684(119888)

and 119901lowast1198684(119888)

are all independent of 119879 and 120591

4 Conclusions and Managerial Implication

In this study we consider the double function of advertisingon (i) buying intention enhancement and (ii) long-termbrand equity building We develop the model under whicha luxury fashion brand will suffer a loss when the advertisingeffort to different customer groups is not up to a certain levelWe then derive the mechanism for the company to identifythe optimal strategies anddecisions in pricing and advertisingbudget allocation

We summarize the major findings related to the optimaladvertising strategies in the following First we find thatwhen there is no penalty for insufficient advertising it isoptimal to allocate the advertising effort to a single customergroup only (ie either LG or FG) When there is a penaltyfor insufficient advertising it is more likely for the optimalpolicy to have the advertising effort allocated to both LG andFG This implies that in light of the penalty for insufficientadvertising the company should take a balance when allocat-ing the advertising effort between the two groups and should

avoid being ldquopolarizedrdquo Second we observe that the optimaladvertising effort is never decreasing in the optimal retailprice when there is no penalty for insufficient advertisingHowever this may not hold when there is penalty for insuf-ficient advertising (for Strategy 2(d) with 119887 gt 1 and Strategy4(c) with 119887 gt 1) Third although there are 14 substrategiesin total we observe that some of them can be screened outby checking carefully their respective necessary conditionsFor example substrategies 2(b) 2(c) 3(c) and 3(d) can bescreened out when 119873

119868ge 0 whilst substrategies 1(b) 2(b)

and 2(c) can be screened out when 119873119868lt 0 Therefore when

determining the global optimal advertising and pricing pol-icy the company should check the value of119873

119868first and then

screen out substrategies that have no locally optimal andfeasible solutions Such screening can significantly reduce thecomputational effort required to solve the problem

For future research it will be important to expand thescope to explore in a supply chain context with issues such assupply chain coordination via incentive alignment schemes(see [24ndash29]) It will also be interesting to explore howcomputerized information systems support the developmentof luxury fashion brands [30ndash32] Last but not least it ispromising to examine how the traditional logistical oper-ational models such as cross-docking [33] and inventoryplanning [34] may be employed to enhance the operationalefficiency of luxury fashion brands

Appendix

Proofs

Proof of Proposition 2 First of all 119890 ge 119879 + 120591 holds underStrategy 1 For 119890 = 119879 + 120591 120582 = 119879(119879 + 120591) is the only feasiblesolution for Strategy 1Therefore 120582lowast = 119879(119879+120591) for 119890 = 119879+120591under Strategy 1 For 119890 gt 119879+120591 by taking the first-order partialderivative of (5) with respect to 120582 we have 120597120587

119871119871(120596)120597120582 =

119890119873119868(119901 minus 119888)(1 + 119887120573) Therefore for any given 119901 gt 119888 and

119890 gt 119879 + 120591 if 119873119868ge 0 then 120587

119871119871(120596) is increasing in 120582 So

120582lowast= Max120582 120582119890 ge 119879 and (1 minus 120582)119890 ge 120591 or 120582lowast = Max120582

120582 ge 119879119890 and 120582 le 1 minus 120591119890 = 1 minus 120591119890 In other words wehave (1 minus 120582lowast)119890 = 120591 for119873

119868gt 0 On the other hand if119873

119868le 0

then 120587119871119871(120596) is decreasing in 120582 So 120582

lowast= Min120582 120582119890 ge

119879 and (1 minus 120582)119890 ge 120591 or 120582lowast = Min120582 120582 ge 119879119890 and 120582 le

1 minus 120591119890 = 119879119890 Hence we have 120582lowast119890 = 119879 for 119873119868lt 0 For

119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587

119871119871(120596) is constant for all 120582 Thus

120582lowast takes any value that satisfy (1minus120582lowast)119890 ge 120591 and 120582lowast119890 le 119879

Proof of Proposition 3 As the feasible set of 120596 for Strategy1(a) is an open set the local optimum for Strategy 1(a) is aninterior point if it exists (i) Clearly 119863

119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast

1198681(119886)) gt 0 are basic conditions for Strategy 1 (ii) By

taking the first- and second-order partial derivatives of (6)with respect to 119901 we have 120597120587

119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus2119866where119866 = 119866(1+ 119887120573) Therefore 120587

119871119871(120596) is strictly concave in

119901 if and only if 119866 gt 0 From the first-order optimal-ity condition 120587

119871119871(120596) is uniquely maximized at 119901lowast

1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)

gt 119888 we obtain 119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591 gt 0Finally by putting 120596lowast

1198681(119886)into (6) we obtain (8)

Proof of Proposition 4 For Strategy 1(b) by taking the first-order and second-order partial derivatives of (9) with respectto 119890 we obtain 120597120587

119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890

2= minus2ℎ lt 0Therefore 120597120587

119871119871(120596) is a concave

function of 119890 Then by considering 120597120587119871119871(120596)120597119890 = 0 we

obtain (10) As 119886(1 + 120573)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119887)

(119901) is strictlyincreasing in 119901

Proof of Proposition 5 As the feasible set of 120596 for Strategy1(b) is an open set the local optimum for Strategy 1(b) isan interior point solution if it exists 119873

119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast

119868119894(119887)) gt 0 are the basic conditions for Strat-

egy 1(b) By putting (10) into (9) we obtain 120587119871119871(120596) =

minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for

Strategy 1(b) we have 120597120587119871119871(120596)120597119901 = minus119884(119901minus119888)2ℎ(1 + 119887120573)

2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587

119871119871(120596) is strictly concave in 119901 if and only if 119884 gt 0

From the first-order optimality condition 120587119871119871(120596) is uniquely

maximized at 119901lowast1198681(119887)

Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591

Then by putting 119901lowast1198681(119887)

into (10) we obtain 119890lowast

1198681(119887) By con-

sidering 119890lowast1198681(119887)

gt 119879 + 120591 and 119884 gt 0 we obtain item (iii) ofProposition 5 which also covers 119861 gt 119873

119868120591 Lastly by putting

120596lowast

1198681(119887)into (9) we obtain (12)

Proof of Proposition 6 By taking the first-order and second-order partial derivatives of (13) with respect to 119890 we obtain120597120587119871119871(120596)120597119890 = 120572(119901 minus 119888)(1 minus 119887)(1 + 119887120573) minus 2ℎ119890 and 1205972120587

119871119871(120596)

1205971198902= minus2ℎ lt 0 Therefore for Strategy 1(c) 120587

119871119871(120596) is a con-

cave function of 119890 and hence the optimal advertising effortsas a function of retail price 119901 is given by (14) As 120572(1 minus

119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)

(119901) is strictly increasing in 119901As the feasible set of 120596 for Strategy 1(c) is an open set the

local optimum for Strategy 1(c) is an interior point solutionif it exists119873

119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic

conditions for Strategy 1(c) By putting (14) into (13) weobtain 120587

119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus

119888)(1+119887120573) andwe have 120597120587119871119871(120596)120597119901 = minus119885(119901minus119888)2ℎ(1+119887120573)

2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587

119871119871(120596) is strictly concave in 119901 if and only if

119885 gt 0 By the first-order optimality condition 120587119871119871(120596) is

uniquely maximized at 119901lowast1198681(119888)

Since 119901lowast1198681(119888)

gt 119888 as 119885 gt 0 wehave 119861 gt minus119873

119868119879 Then by putting 119901lowast

1198681(119888)into (14) we obtain

119890lowast

1198681(119888) By considering 119890lowast

1198681(119888)gt 119879 + 120591 we obtain item (iii)

of Proposition 6 which also covers 119861 gt minus119873119868119879 Finally by

putting 120596lowast1198681(119888)

into (13) we obtain (16)

Proof of Proposition 7 First of all 119890 ge 119879 for Strategy 2 For 119890 =119879 120582 = 1 is the only feasible solution for Strategy 2Thereforethe optimal value of 120582 for Strategy 2 with 119890 = 119879 is 120582lowast = 1 For119890 gt 119879 by taking the first-order partial derivative of (17) withrespect to 120582 we have 120597120587

119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890

for Strategy 2 Therefore for any given 119901 gt 119888 and 119890 gt 119879 if119873119868(119901 minus 119888) gt 120583(1 + 119887120573) then 120587

119871119871(120596) is increasing in 120582 As

120582 = 1 satisfies 120582119890 ge 119879 and (1minus120582)119890 lt 120591 120582lowast = 1 for119873119868(119901minus 119888) gt

120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing

in 120582 Therefore for 119879 lt 119890 lt 119879 + 120591 and119873119868(119901 minus 119888) lt 120583(1 + 119887120573)

the optimal 120582lowast for Strategy 2 satisfies 120582lowast119890 = 119879 Next let 1205961015840 =(119890 1205821015840 119901) and 12059610158401015840 = (119890 12058210158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888119873

119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863

119865(12059610158401015840) gt 0 1205821015840119890 ge 119879 12058210158401015840119890 ge 119879 (1minus1205821015840)119890 lt 120591 and

(1minus12058210158401015840) = 120591 We have 120587

119871119871(1205961015840) lt 120587119871119871(12059610158401015840) Therefore Strategy

1 dominates Strategy 2 for 119890 ge 119879+120591 and119873119868(119901minus119888) lt 120583(1+119887120573)

If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587

119871119871(120596) remains constant for

all values of 120582 Therefore any 120582 that satisfies (1 minus 120582)119890 lt 120591 and120582119890 ge 119879 is an optimal solution As there exist multiple 120582whichsatisfy (1minus120582)119890 lt 120591 and 120582119890 ge 119879 there exist multiple 120582lowast for thecase119873

119868(119901 minus 119888) = 120583(1 + 119887120573)

Proof of Proposition 8 The local optimum for Strategy 2(a)is an interior point solution if it exists 119863

119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 are the basic conditions for Strategy 2 in turn

they are necessary conditions for Strategy 2(a) tooBy takingthe first- and second-order partial derivatives of (18) withrespect to 119901 we have 120597120587

119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is

defined in the proof of Proposition 3 Therefore 120587119871119871(120596) is

strictly concave in 119901 if and only if 119866 gt 0 By the first-orderoptimality condition 120587


1198682(119886)

Since 119901lowast1198682(119886)

gt 119888 we obtain item (iii) of Proposition 8 Lastlyby putting 120596lowast

1198682(119886)into (18) we obtain (20)

Proof of Proposition 9 By taking the first-order and second-order partial derivatives of (21) with respect to 119890 we obtain120597120587119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890 and 1205972120587

119871119871(120596)


119871119871(120596) is a concave function of 119890 for Strategy

2(b) Hence the optimal advertising effort as a function ofretail price 119901 is given by (22) Finally by direct observation119890lowast

1198682(119887)(119901) is strictly increasing in 119901

Similarly the local optimum for Strategy 2(b) is an inte-rior point if it exists Clearly conditions119873

119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast

1198682(119887)) gt 0 are necessary for Strategy 2(b) By

putting (22) into (21) we obtain 120587119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 +

119887120573)2+ 119861(119901 minus 119888)(1 + 119887120573) minus 120583120591 and we have 120597120587

119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2

Therefore120587119871119871(120596) is strictly concave in119901 if and only if119884 gt

0 By the first-order optimality condition 120587119871119871(120596) is uniquely


As 119901 gt 119888 we obtain 119861 gt 0 By putting119901lowast

1198682(119887)into (22) we obtain 119890lowast

1198682(119887) As 119890 gt 119879 for Strategy 2(b)

we need 119861 gt 119884119879[119886(1 + 120573)] gt 0 which covers the conditionfor119901 gt 119888 By considering119873

119868(119901lowast

1198682(119887)minus119888) gt 120583(1+119887120573) we obtain

item (iv) of Proposition 9 Lastly by putting 120596lowast1198682(119887)

into (21)we obtain (24)

Proof of Proposition 10 Similarly the local optimum forStrategy 1(c) is an interior point if it exists Conditions119873119868gt 0 119863

119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast

1198682(119888)) gt 0 are necessary for

Strategy 2(c) As119873119868(119901 minus 119888) = 120583(1 + 119887120573) for Strategy 2(c) we

obtain 119901lowast1198682(119888)

= 119888 + 120583(1 + 119887120573)119873119868gt 119888 By putting 120596lowast

1198682(119888)into

120587119871119871(120596) we obtain (27) Finally by considering 119890lowast

1198682(119888)gt 119879 we

obtain item (ii) of Proposition 10


Proof of Proposition 11 By taking the first-order and second-order partial derivatives of (28) with respect to 119890 we obtain120597120587119871119871(120596)120597119890 = 120572(119901 minus 119888)(1 minus 119887)(1 + 119887120573) minus 2ℎ119890 + 120583 and

1205972120587119871119871(120596)120597119890

2= minus2ℎ lt 0 Therefore 120597120587

119871119871(120596) is a concave

function of 119890 for Strategy 2(d) and for any given 119901 gt 119888 theoptimal advertising effort as a function of retail price 119901 isgiven by (25)

The local optimum for Strategy 1(d) is an interior pointif it exists 119863

119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic

conditions of Strategy 2 By putting (29) into (28) we obtain120587119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus

119887)](119901minus119888)2ℎ(1+119887120573)minusℎ1205832minus120583[119879+120591minus120583] andwe have 120597120587

119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587


119885 gt 0 From the first-order optimality condition 120587119871119871(120596) is


By considering 119879 lt 119890lowast

1198682(119889)lt

119879 + 120591 we obtain item (iii) of Proposition 11 By considering119901lowast

1198682(119889)gt 119888 we obtain item (iv) of Proposition 11 Then by

considering119873119868(119901lowast

1198682(119889)minus 119888) lt 120583(1 + 119887120573) we obtain item (v) of

Proposition 11 Furthermore by putting 119901lowast1198682(119889)

into (29) weobtain 119890

lowast

1198682(119889) Finally by putting 120596lowast

1198682(119889)into (28) we obtain

(31)

Proof of Proposition 12 First of all 119890 ge 120591 for Strategy 3 For119890 = 120591 120582 = 0 is the only feasible solution Therefore theoptimal value of 120582 for Strategy 3 when 119890 = 120591 is 120582lowast = 0 For119890 gt 120591 by taking the first-order partial derivative of (32) withrespect to 120582 we have 120597120587

119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890

Therefore for any given 119901 gt 119888 and 119890 gt 120591 if 119873119868(119901 minus 119888) gt

minus119898(1 + 119887120573) then 120587119871119871(120596) is strictly increasing in 120582 For 120591 lt

119890 lt 119879+ 120591 120582 = 1 minus 120591119890 is the biggest value that satisfies 120582119890 lt 119879and (1 minus 120582)119890 ge 120591 Therefore 120582lowast = 1 minus 120591119890 for Strategy 3 if119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) and 120591 lt 119890 lt 119879 + 120591 Next consider

1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879

(1 minus 1205821015840)119890 ge 120591 and (1 minus 12058210158401015840) ge 120591 We have 120587

119871119871(1205961015840) lt 120587119871119871(12059610158401015840)

Therefore Strategy 1 dominates Strategy 3 for 119890 ge 119879 + 120591 and119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) If 119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573) then

120587119871119871(120596) is strictly decreasing in120582 As120582 = 0 satisfies120582119890 lt 119879 and

(1minus120582)119890 ge 120591 120582lowast = 0 for Strategy 3 if119873119868(119901minus119888) lt minus119898(1+119887120573) If

119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587

119871119871(120596) is independent of 120582 for

Strategy 3 As there are multiple 120582 which satisfy (1 minus 120582lowast)119890 lt 120591and 120582lowast119890 ge 119879 there exist multiple 120582lowast for the case119873

119868(119901 minus 119888) =

minus119898(1 + 119887120573)

Proof of Proposition 13 Similarly the local optimum forStrategy 3(a) is an interior point if it exists 119863

119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast

1198683(119886)) gt 0 are necessary for any substrategies of

Strategy 3 By taking the first- and second-order partialderivatives of (33) with respect to 119901 we have120597120587119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 + 120572(1 minus 119887)120591])(1 + 119887120573) and

1205972120587119871119871(120596)120597119901

2

= minus2119866 where 119866 is defined in the proof ofProposition 3 Therefore 120587

119871119871(120596) is strictly concave in 119901 if

and only if 119866 gt 0 From the first-order optimality condition120587119871119871(120596) is uniquely maximized at 119901lowast

1198683(119886) As 119901 gt 119888 we obtain

item (iii) of Proposition 13 Finally by putting 120596lowast

1198683(119886)into

(33) we obtain (34)

Proof of Proposition 14 By taking the first-order and second-order partial derivatives of (36) with respect to 119890 we obtain120597120587119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890 + 119898 and

1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave

function of 119890 for Strategy 3(b) and for any given 119901 gt 119888 Hencethe optimal advertising effort as a function of retail price 119901 isgiven by (37) Next as 119886(1 + 120573)[2ℎ(1 + 119887120573)] gt 0 119890lowast

1198683(119887)(119901) is

strictly increasing in 119901Similarly the local optimum for Strategy 1(a) is an interior

point if it exists 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the

basic conditions of Strategy 3 By putting (37) into (36) weobtain 120587

119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)](119901 minus 119888)2ℎ(1 + 119887120573) minus 119898(119879 + 120591) + 11989824ℎ and we

have 120597120587119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2

Therefore 120587119871119871(120596) is strictly concave in 119901 if and only if 119884 gt 0



By considering 119901lowast1198683(119887)

gt 119888 we obtainitem (iii) of Proposition 14 Similarly by considering 120591 lt

119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast

1198683(119887)minus 119888) gt minus119898(1 + 119887120573) we obtain

items (iv) and (v) of Proposition 14 respectively By putting119901lowast


1198683(119887) Finally by putting (38) into

(36) we obtain (39)


119871119871(120596)

1205971198902= minus2ℎ lt 0 Therefore for any given 119901 gt 119888 120587

119871119871(120596) is

a concave function of 119901 under Strategy 3(c) Therefore theoptimal advertising effort as a function of retail price 119901 isgiven by (41) As 120572(1minus119887)[2ℎ(1+119887120573)] gt 0 119890lowast

1198683(119888)(119901) is strictly

increasing in 119901Similarly the local optimum for Strategy 3(c) is an inte-

rior point if it exists119873119868lt 0119863

119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt

0 are the basic conditions for Strategy 3 By putting (41) into(40) we obtain 120587

119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus

119888)(1 + 119887120573) minus 119898119879 and we have 120597120587119871119871(120596)120597119901 = minus119885(119901 minus

119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587





gt 119888 weobtain 119861 gt 0 By considering 119890lowast

1198683(119888)gt 120591 we obtain item (iii) of

Proposition 15 By considering119873119868(119901lowast

1198683(119888)minus 119888) lt minus119898(1 + 119887120573)

we obtain 119861 ge minus119898119885(2ℎ119873119868) gt 0 Finally by putting 120596lowast

1198683(119888)


Proof of Proposition 16 Similarly the local optimum forStrategy 3(d) is an interior point if it exists 119873

119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 are specific conditions of

Strategy 3(d) As 119873119868(119901 minus 119888) = minus119898(1 + 119887120573) for Strategy 3(d)

we obtain 119901lowast

1198683(119889) Notice that 119901lowast

1198683(119889)gt 0 as 119873

119868lt 0 for

Strategy 3(d) According to Proposition 23 we obtain119890lowast

1198683(119889)= 119890lowast

1198683(119889)(119901lowast

1198683(119889)) By putting 120596lowast

1198683(119889)into (17) we obtain

(46) Finally by considering 119890lowast1198683(119889)

gt 120591 we obtain item (ii) ofProposition 16


Proof of Proposition 17 Consider 1205961015840 = (1198901015840 1205821015840 1199011015840) and 12059610158401015840 =

(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt

0 and 119863119865(12059610158401015840) gt 0 By following the proof of Proposition 12

for any fixed 119901 gt 119888 and 119879 le 119890 lt 119879 + 120591 120587119871119871(120596) is strictly

increasing in 120582 Therefore 120587119871119871(12059610158401015840) gt 120587

119871119871(1205961015840) However

12059610158401015840= (1198901015840 12058210158401015840 1199011015840) is a solution of Strategy 2 instead of Strategy

4 for 12058210158401015840 rarr 1 Hence the optimal 120582lowast for Strategy 4 does notexist (ie Strategy 2 dominates Strategy 4 for any fixed 119901 gt 119888and 119879 le 119890 lt 119879 + 120591) This completes the proof of part (a) Theproof of part (b) and part (c) are similar to the proof of part(a)

Proof of Proposition 18 First of all 119890 lt 119879 + 120591 for Strategy 4 If119890 = 0 then 120582 can be ignored For 0 lt 119890 lt 119879 + 120591 by taking thefirst-order partial derivative of (47) with respect to 120582 we have120597120587119871119871(120596)120597120582 = 119873

119868119890(119901minus119888)(1+119887120573)+(119898minus120583)119890Therefore for any

fixed 119901 gt 119888 and 0 lt 119890 lt 119879 (a) if119873119868(119901 minus 119888) gt (120583 minus 119898)(1 + 119887120573)

then 120587119871119871(120596) is increasing in 120582 Hence the optimal value of 120582

in this case is 120582lowast = 1 (please note that 120582lowast = 1 satisfies 120582119890 lt 119879and (1 minus 120582)119890 lt 120591 for 0 lt 119890 lt 119879) (b) if 119873

119868(119901 minus 119888) lt (120583 minus

119898)(1+119887120573) then 120587119871119871(120596) is decreasing in 120582 Hence the optimal

value of 120582 in this case is 120582lowast = 0 (please note that 120582lowast = 0

satisfies 120582119890 lt 119879 and (1 minus 120582)119890 lt 120591 for 0 lt 119890 lt 120591) (c) if119873119868(119901 minus

119888) = (120583minus119898)(1+119887120573) then120587119871119871(120596) is independent of 120582 As there

are multiple 120582 which satisfy (1 minus 120582lowast)119890 lt 120591 and 120582lowast119890 lt 119879 for0 lt 119890 lt 119879+120591 there exists multiple 120582lowast for the case119873

119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)

Proof of Proposition 19 Similarly the local optimum forStrategy 4(a) is an interior point if it exists Conditions119863119871(120596lowast

1198684(119886)) gt 0 and 119863

119865(120596lowast

1198684(119886)) gt 0 are necessary for any

substrategies of Strategy 4 By taking the first- and second-order partial derivatives of (48) with respect to 119901 we have120597120587119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + 119861)(1 + 119887120573) and 120597

2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587

119871119871(120596) is strictly concave in 119901 if and

only if 119866 gt 0 From the first-order optimality condition120587119871119871(120596) is uniquely maximized at 119901lowast

1198684(119886) By considering

119901lowast

1198684(119886)gt 119888 we obtain 119861 gt 0 By putting 120596lowast

1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave

function of 119890 for Strategy 4(b) Thus for any given 119901 gt 119888the optimal advertising effort as a function of retail price119901 is given by (52) According to Proposition 16 119890lowast

1198683(119887)(119901) is

strictly increasing in 119901 Therefore 119890lowast1198684(119887)

(119901) which is equal to119890lowast

1198683(119887)(119901) is also strictly increasing in 119901

Similarly the local optimum for Strategy 4(b) is aninterior point if it exists Conditions 119863

119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast

1198684(119887)) gt 0 are necessary conditions for Strategy 4(b)

By putting (52) into (51) we obtain 120587119871119871(120596) = minus119884(119901 minus

119888)24ℎ(1 + 119887120573)

2+[2ℎ119861+119886119898(1+120573)](119901minus119888)2ℎ(1+119887120573)minus119898119879minus120583120591+

11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2

Therefore 120587119871119871(120596) is strictly concave in 119901 if and only if




gt 119888 weobtain 2ℎ119861 + 119886119898(1 + 120573) gt 0 Then by putting 119901lowast

1198684(119887)into

(52) we obtain 119890lowast1198684(119887)

By considering 119873119868(119901lowast

1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast

1198684(119887)lt 119879 we obtain item (iv) and item

(v) of Proposition 20 respectively Finally by putting 120596lowast1198684(119887)



1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave

function of 119890 for Strategy 4(c) for any given 119901 gt 119888 Fromthe first-order optimality condition the optimal advertisingeffort as a function of retail price 119901 is given by (50)


1198684(119888)) gt 0 and 119863

119865(120596lowast


Strategy 4(c) By putting (56) into (55) we obtain 120587119871119871(120596) =

minus119885(119901 minus 119888)24ℎ(1 + 119887120573)

2+[2ℎ119861+120572120583(1minus119887)](119901minus119888)2ℎ(1+119887120573)minus

119898119879 minus 120583120591 + 12058324ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2





gt 119888 weobtain item (ii) of Proposition 22 By considering 0 lt 119890lowast

1198684(119888)lt

120591 and 119873119868(119901lowast

1198684(119888)minus 119888) lt (120583 minus 119898)(1 + 119887120573) we obtain item (iv)

and item (v) of Proposition 22 respectively Finally by putting120596lowast

1198684(119888)into (55) we obtain (56)

Acknowledgement

This paper is supported by the ldquo985 Projectrdquo of Sun Yat-SenUniversity (Account nos 14000-3282101 and 14000-3281303)The authors sincerely thank the anonymous reviewers fortheir constructive comments on this paper

References

[1] T M Choi S C Liu K M Pang and P S Chow ldquoShoppingbehaviors of individual tourists from the Chinese Mainland toHong Kongrdquo Tourism Management vol 29 no 4 pp 811ndash8202008

[2] S C Liu TMChoi R Au andC LHui ldquoA study on individualtourists from theChinesemainland toHongKong implicationsfor tourism marketing in fashionrdquo Tourism Economics vol 17pp 1287ndash1309 2011

[3] T M Choi S C Liu C S Tang and Y Yu ldquoA cross-cluster andcross-region analysis of fashion brand extensionsrdquo The Journalof the Textile Institute vol 102 no 10 pp 890ndash904 2011

[4] T M Choi P S Chow and T Xiao ldquoElectronic price-testingscheme for fashion retailing with information updatingrdquo Inter-national Journal of Production Economics vol 140 pp 396ndash4062012

[5] D A Aaker and A L Biel Brand Equity amp Advertising Adver-tisingrsquos Role in Building Strong Brands Lawrence Erlbaum 1993


[6] T L Childers and A R Rao ldquoThe influence of familial andpeer-based reference groups on consumer decisionsrdquo Journal ofConsumer Research vol 19 no 2 pp 198ndash211 1992

[7] M Grinblatt M Keloharju and S Ikaheimo ldquoSocial influenceand consumption evidence from the automobile purchases ofneighborsrdquo Review of Economics and Statistics vol 90 no 4 pp735ndash753 2008

[8] M B Brewer ldquoThe social self on being the same and different atthe same timerdquo Personality and Social Psychology Bulletin vol17 no 5 pp 475ndash482 1991

[9] K T Tian W O Bearden and G L Hunter ldquoConsumersrsquo needfor uniqueness scale development and validationrdquo Journal ofConsumer Research vol 28 no 1 pp 50ndash66 2001

[10] A M Fionda and C M Moore ldquoThe anatomy of the luxuryfashion brandrdquo Journal of Brand Management vol 16 no 5-6pp 347ndash363 2009

[11] K L Keller ldquoManaging the growth tradeoff challenges andopportunities in luxury brandingrdquo Journal of Brand Manage-ment vol 16 no 5-6 pp 290ndash301 2009

[12] P Bourdieu and R Nice Distinction A Social Critique of theJudgment of Taste Harvard University Press 1984

[13] B Bryson ldquolsquoAnything but heavy metalrsquo symbolic exclusion andmusical dislikesrdquo American Sociological Review vol 61 no 5pp 884ndash899 1996

[14] J Zheng C H Chiu and T M Choi ldquoOptimal advertising andpricing strategies for luxury fashion brands with social influ-encesrdquo IEEE Transactions on Systems Man and CyberneticsPart A vol 42 pp 827ndash837 2012

[15] D Krahmer ldquoAdvertising and conspicuous consumptionrdquo Jour-nal of Institutional andTheoretical Economics vol 162 no 4 pp661ndash682 2006

[16] L Grosset and B Viscolani ldquoOptimal dynamic advertising withan adverse exogenous effect on brand goodwillrdquo Automaticavol 45 no 4 pp 863ndash870 2009

[17] B Ghosh and A Stock ldquoAdvertising effectiveness digital videorecorders and productmarket competitionrdquoMarketing Sciencevol 29 no 4 pp 639ndash649 2010

[18] B N Anand and R Shachar ldquoTargeted advertising as a signalrdquoQuantitativeMarketing and Economics vol 7 no 3 pp 237ndash2662009

[19] H Raghavan and R Iyer Probabilistic First Pass Retrieval forSearch Advertising fromTheory to Practice ACM 2010

[20] J McClure and E Kumcu ldquoPromotions and product pricingparsimony versus Veblenesque demandrdquo Journal of EconomicBehavior and Organization vol 65 no 1 pp 105ndash117 2008

[21] K Yamada ldquoMacroeconomic implications of conspicuous con-sumption a Sombartian dynamic modelrdquo Journal of EconomicBehavior and Organization vol 67 no 1 pp 322ndash337 2008

[22] C H Chiu J Zheng and T M Choi ldquoOptimal pricing andinventory decisions for fashion retailers under value-at-riskobjectiverdquo in Fashion Supply Chain Management Industry andBusiness Analysis T M Choi Ed chapter 5 pp 100ndash109 IGIGlobal 2011

[23] CMarinelli ldquoThe stochastic goodwill problemrdquo European Jour-nal of Operational Research vol 176 no 1 pp 389ndash404 2007

[24] B Shen TM Choi YWang andC K Y Lo ldquoThe coordinationof fashion supply chains with a risk averse supplier by themarkdown money policyrdquo IEEE Transactions on Systems Manand Cybernetics 2013

[25] N Liu T M Choi M C W Yuen and F Ng ldquoOptimal pricingmodularity and return policy under mass customizationrdquo IEEE

Transactions on Systems Man and Cybernetics Part A vol 42pp 604ndash614 2012

[26] C H Chiu T M Choi and C S Tang ldquoPrice rebate andreturns supply contracts for coordinating supply chains withprice-dependent demandsrdquo Production and Operations Man-agement vol 20 no 1 pp 81ndash91 2011

[27] C H Chiu T M Choi H T Yeung and Y Zhao ldquoSales rebatecontracts in fashion supply chainsrdquo Mathematical Problems inEngineering vol 2012 Article ID 908408 19 pages 2012

[28] H J Peng and M H Zhou ldquoQuantity discount supply chainmodels with fashion products and uncertain yieldrdquoMathemat-ical Problems in Engineering In press

[29] S Costantini G De Gasperis A Provetti and P TsintzaldquoHeuristic approach to proposal-based negotiationrdquoMathemat-ical Problems in Engineering In press

[30] TM Choi and S Sethi ldquoInnovative quick response programs areviewrdquo International Journal of Production Economics vol 127no 1 pp 1ndash12 2010

[31] TMChoi P S Chow and S C Liu ldquoImplementation of fashionERP systems inChina case study review and future challengesrdquoInternational Journal of Production Economics 2013

[32] T M Choi C L Hui S F Ng and Y Yu ldquoColor trend forecast-ing of fashionable products with very few historical datardquo IEEETransactions on Systems Man and Cybernetics Part C vol 42pp 1003ndash1010 2012

[33] Z H Hu Y X Zhao and T M Choi ldquoVehicle routing problemfor fashion supply chains with cross-dockingrdquo MathematicalProblems in Engineering In press

[34] T M Choi and C H Chiu ldquoMean-downside-risk and mean-variance newsvendor models implications for sustainable fash-ion retailersrdquo International Journal of Production Economics vol135 pp 552ndash560 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


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Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


group (LG) and the follower group (FG) respectively Theformer group usually plays a leadership role in the fashiontrend whereas the latter one is usually following the purchaseof LG

In light of the different buying behaviors between the twogroups of consumers luxury fashion brands must target eachof them effectively and efficiently with proper allocation thelimited advertising budget in order to obtain its best payoffTake the classic high-end British fashion brand Burberryas an example Burberry holds its own fashion shows inMilan every year to attract its elite customers The brand alsostrengthens the social interaction with its mass customersit is now the leading luxury fashion brand on Facebookwith over one million fans In short Burberry utilizes thesocial influence for brand development establishing a leadingpresence across social media platforms and creating newcommunities of interest

Zheng et al [14] explore the impact of social influences onthe optimal pricing and advertising allocation strategies of aluxury fashion brand in the presence of LG and FG whichhave social influences on each other In this paper we extendZheng et al [14] with more realistic considerations To bespecific we investigate how a luxury fashion brand allocatesthe limited advertising expense strategically to maximize itsprofit Similar to Zheng et al [14] we consider a retail brandowner that sells a luxury fashion product to the market thatconsists of two groups of consumers (LG and FG) who havesocial influence on each other We consider the scenario thatthere is a certain ldquobasic amountrdquo of advertising effort for eachmarket segment (FG LG) in order to maintain the brandstrength otherwise the brand would suffer in the long rundue to loss of goodwill from a group of consumers To reflectthat a brand has concerns on its budget allocation for sus-taining the respective brandrsquos rolepositioning in the marketwith respect to both groups of consumers we consider thatthere is a penalty cost in the form of a linear loss function forany advertising effort lower than the basic amount for eachmarket segment

2 Literature Review

Inmarketing science and operationsmanagement there are aconsiderable amount of studies that examine optimization ofanalytical models with pricing andor advertising decisionsunder different settings To take into consideration of theeffect of advertising Krahmer [15] considers a model withadvertising that informs the public of brand names and cre-ates the possibility of conspicuous consumption by renderingbrands as a signaling device The author shows that advertis-ing increases consumersrsquo willingness to pay which providesa foundation based on optimization behavior for persuasiveapproaches to advertising Grosset andViscolani [16] proposea model of a firm that advertises a product in a homoge-neous market where a constant exogenous interference ispresent They reveal that the optimal policy takes one of twoforms either a positive and constant advertising effort or adecreasing effort starting from a positive level and eventuallyreaching the zero value at a finite exit time Ghosh and Stock[17] use a model of informative advertising to study the

effect of penetration on competing advertisersrsquo strategies andprofits Conditions under which an increase in penetrationcounter intuitively leads firms to increase advertising levelsand enjoy higher profits are identified In Zheng et al [14] theauthors consider that advertising has a direct impact on socialneeds but lack of considering the impact of the penalty of theinsufficient advertising on the target groups It is importantto consider the loss from the insufficient advertising forthe target group because in practices low advertising efforton the market might have a negative impact on marketdemand

It is well known that advertising on target groups canincrease the grouprsquos consumption In practice most of thetime advertising is targeted [18 19] Anand and Shachar [18]study a model in which firms can target their advertisementsto particular groups of consumers and advertising is noisyThey consider the case that a particular product has a betterfit with the tastes of some consumers than those of the oth-ers and consumer-utility depends on the resulting ldquomatchrdquobetween product attributes and their tastes Raghavan andIyer [19] examine advertising strategy when competing firmscan target their advertising effort to different groups of con-sumers within a market With targeted advertising they findthat firms advertise more to consumers who have a strongpreference for their product than to comparison shopperswho can be attracted to the competition They argue thatadvertising less on comparison shoppers can be seen as away for firms to endogenously increase differentiation in themarket Interestingly they also find that target advertisingleads to higher profits regardless of the firmsrsquo ability to settargeted prices

Our study is related to the work in social factors and con-spicuous consumption [15 20 21] Focusing on the conspic-uous products Krahmer [15] indicates that brands are con-sumed for image reasons and advertising creates a brandrsquosimage He argues that advertising informs the public of brandnames and creates the possibility of conspicuous consump-tion In a price-competition framework he shows that adver-tising increases consumersrsquo willingness to pay and thus pro-vides a foundation for determining the optimal advertisingstrategyMoreover he finds that an incumbentmight strategi-cally overinvest in advertising to deter entry and competitionmight be socially undesirable McClure and Kumcu [20]formalize the relationship between the optimal pricequantitycombination and the thoroughness of conspicuous productpromotionsThey reveal that iterating towards the profitmax-imizing thoroughness of product promotion will lead to abackward bending pricequantity locus In this paper westudy an optimal control problem in the fashion apparelindustryWe establish the studywhich analytically shows howa firm allocates the target advertising strategically on the lux-ury fashion brands when facing the market demand isaffected by consumer desire for exclusivity and conformityrespectively

3 The Model

We consider the scenario that a company sells a fashionproduct to the market with two groups of customers namely







119863119871(120596) = [119881

119871(120596)]+

119863119865(120596) = [119881

119865(120596)]+

(1)

where 119881119871(120596) = 119909

119871+ 119886120582119890 minus 119887119863

119865(120596) minus 119892119901 119881

119865(120596) = 119909

119865+ 119886(1 minus

120582)119890 + 120573119863119871(120596) minus 120574119901 [119884]+ = max0 119884 and 119909

119871 119909119865 119886 120572 119887



119871(119890) = 119898[119879 minus 120582119890]

+ and Λ119865(119890) = 120583[120591 minus




119871(119890) minus Λ

119865(119890)


+


(2)


max120596isinΩ


119871(119890) minus Λ

119865(119890)

(P-SILL)






119871119871(120596) gt 0



119865gt 120574119888













119871(120596) gt 0 and 119863







119863 (120596) =


(1 + 119887120573)

(3)


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

(1 + 119887120573)



+

(4)


120573) minus 120572(1 minus 119887)


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573

minus ℎ1198902

(5)






119868= 0



119868first If



119868ge 0

If 119873119868le 0 then 120582

lowast119890 = 119879 and (1 minus 120582



and 120582lowast119890 ge 119879 if 119873










119868= 0





119868le 0 and 119890lowast gt 119879 + 120591


119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

1 + 119887120573

+

(119901 minus 119888) [119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]

1 + 119887120573

minus ℎ(119879 + 120591)2

(6)



119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198681(119886)= 119890lowast

1198681(119886)= 119879 + 120591 120582

lowast

1198681(119886)=

119879

(119879 + 120591)

119901lowast

1198681(119886)=

119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591

2119866

+ 119888

(7)

120587119871119871(120596lowast

1198681(119886)) =

[119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]2

4119866 (1 + 119887120573)

minus ℎ(119879 + 120591)2

(8)






120582lowast1198681(119886)

and119901lowast


1198681(119886) 120582lowast1198681(119886)

and 119901lowast



119889119879 = 1 gt 0 119889120582lowast1198681(119886)

119889119879 =

120591(119879 + 120591)2gt 0 and119889119901lowast

1198681(119886)119889119879 = 119886(1+120573)(2119866) gt 0) In other



and 120582lowast1198681(119886)






1198681(119886)and

120587119871119871(120596lowast




119865(120596) gt 0

120582lowast= 1 minus 120591119890



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573

minus ℎ1198902

(9)


119890lowast

1198681(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) (10)

and 119890lowast1198681(119887)





119868ge 0 119863

119871(120596lowast

1198681(119887)) gt 0 and 119863

119865(120596lowast

1198681(119887)) gt 0



120596lowast

1198681(119887)= 119890lowast

1198681(119887) 120582lowast

1198681(119887) 119901lowast

1198681(119887) (11)

120587119871119871(120596lowast

1198681(119887)) =

ℎ(119861 minus 119873119868120591)2

119884

(12)

where 119890lowast1198681(119887)

= 119886(1 + 120573)(119861 minus 119873119868120591)119884 120582lowast

1198681(119887)= 1 minus 120591119890

lowast

1198681(119887)

and 119901lowast119868119894(119887)

= 119888 + 2ℎ(119861 minus 119873119868120591)(1 + 119887120573)119884



1198681(119887)gt 119888 if and only

if 119861 gt 119873119868120591 Because 119861 gt 119884(119879 + 120591)[119886(1 + 120573)] + 119873

119868120591 implies




120582lowast1198681(119886)

119901lowast1198681(119886)

and120587119871119871(120596lowast


lowast

1198681(119886)119889119879 = minus119884119861119886(1 +



1198681(119886) 120582lowast1198681(119886)

119901lowast

1198681(119886) and 120587

119871119871(120596lowast

1198681(119887))


1198681(119887)and 120587

119871119871(120596lowast




119865(120596) gt 0





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573

minus ℎ1198902

(13)





119890lowast

1198681(119888)(119901) =

120572 (1 minus 119887)

2ℎ (1 + 119887120573)

(119901 minus 119888) (14)

and 119890lowast1198681(119888)


only if (i) 119873119868le 0 119863

119871(120596lowast

1198681(119888)) gt 0 and 119863

119865(120596lowast

1198681(119888)) gt 0 (ii)



120596lowast

1198681(119888)= 119890lowast

1198681(119888) 120582lowast

1198681(119888) 119901lowast

1198681(119888) (15)

120587119871119871(120596lowast

1198681(119888)) =

ℎ(119861 + 119873119868119879)2

119885

(16)

where 119890lowast1198681(119888)

= 120572(1minus119887)(119861+119873119868119879)119885 120582lowast

1198681(119888)= 119879119885120572(1minus119887)(119861+

119873119868119879) and 119901lowast

1198681(119888)= 2ℎ(119861 + 119873

119868119879)(1 + 119887120573)119885 + 119888




119868119894(119888)gt 119888 if




1198681(119888)) 119890lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888) 120582lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888)and 120587

119871119871(120596lowast




120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(17)


119868(119901 minus 119888) lt 120583(1 + 119887120573) and


119868(119901 minus 119888) lt





119868(119901minus119888) gt 120583(1+119887120573)







119868(119901 minus 119888) = 120583(1 + 119887120573) then the







and


and 120582lowast = 119879119890


119871(120596) gt 0

and119863119865(120596) gt 0

Denote by 120596lowast1198682(119894)



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119879]

1 + 119887120573

minus ℎ1198792minus 120583120591

(18)



119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 + 119886(1 + 120573)119879 gt 0


120596lowast

1198682(119886)= 119890lowast

1198682(119886)= 119879 120582

lowast

1198682(119886)= 1

119901lowast

1198682(119886)= 119888 +

[119861 + 119886 (1 + 120573) 119879]

(2119866)

(19)

120587119871119871(120596lowast

1198682(119886)) =

1198612+ 2119861119886 (1 + 120573) 119879 minus 119884119879

2

4119866 (1 + 119887120573)

minus 120583120591 (20)



1198682(119886)gt 119888 Moreover


1198682(119886)and

119901lowast



1198682(119886)and 119901lowast

1198682(119886)are



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 120583120591

(21)


119873119868gt 0


119890lowast

1198682(119887)(119901) = 119890

lowast

1198681(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(22)

and 119890lowast1198682(119887)


only if (i) 119873119868gt 0 119863

119871(120596lowast

1198682(119887)) gt 0 and 119863

119865(120596lowast

1198682(119887)) gt 0 (ii)



120596lowast

1198682(119887)= 119890lowast

1198682(119887) 120582lowast

1198682(119887) 119901lowast

1198682(119887) (23)

120587119871119871(120596lowast

1198682(119887)) =

ℎ1198612

119884

minus 120583120591 (24)

where 119890lowast1198682(119887)

= 119886(1 + 120573)119861119884 120582lowast1198682(119887)

= 1 and 119901lowast1198682(119887)

= 2ℎ119861(1 +

119887120573)119884 + 119888





119868(119901 minus 119888) gt 120583(1 + 119887120573) which is




119871119871(120596lowast



1198682(119887) 120582lowast1198682(119887)

and 119901

lowast



1198682(119887) 120582lowast1198682(119887)

and 119901lowast1198682(119887)



119865(120596) gt 0


119868(119901minus 119888) = 120583(1+119887120573) implies

119873119868gt 0


119890lowast

1198682(119888)(119901) = 119890

lowast

1198682(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(25)

and 119890lowast1198682(119888)


only if (i)119873119868gt 0119863

119871(120596lowast

1198682(119888)) gt 0 and119863

119865(120596lowast

1198682(119888)) gt 0 and (ii)



120596lowast

1198682(119888)= 119890lowast

1198682(119888) 120582lowast

1198682(119888) 119901lowast

1198682(119888) (26)

120587119871119871(120596lowast

1198682(119888)) =

119861120583

119873119868

minus

1198841205832

4ℎ1198732

119868

minus 120583120591 (27)

where 119890lowast1198682(119888)

= 119886120583(1 + 120573)(2ℎ119873119868) 120582lowast1198682(119888)

= 1 and 119901lowast1198682(119888)

= 119888 +

120583(1 + 119887120573)119873119868







1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)


1198682(119888)and119901lowast

1198682(119888)are




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573


(28)


119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)


119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873



120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]







1198682(119889)gt 119888 Finally the




and 119901lowast

1198682(119889) we


and 119901lowast1198682(119889)









1198682(119889) 120582lowast1198682(119889)

and 119901lowast



120582lowast1198682(119889)

and 119901lowast1198682(119889)



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(32)



119868(119901 minus 119888) gt minus119898(1 + 119887120573)







119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)









119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119863119871(120596) = [119881

119871(120596)]+

119863119865(120596) = [119881

119865(120596)]+

(1)

where 119881119871(120596) = 119909

119871+ 119886120582119890 minus 119887119863

119865(120596) minus 119892119901 119881

119865(120596) = 119909

119865+ 119886(1 minus

120582)119890 + 120573119863119871(120596) minus 120574119901 [119884]+ = max0 119884 and 119909

119871 119909119865 119886 120572 119887



119871(119890) = 119898[119879 minus 120582119890]

+ and Λ119865(119890) = 120583[120591 minus




119871(119890) minus Λ

119865(119890)


+


(2)


max120596isinΩ


119871(119890) minus Λ

119865(119890)

(P-SILL)






119871119871(120596) gt 0



119865gt 120574119888













119871(120596) gt 0 and 119863







119863 (120596) =


(1 + 119887120573)

(3)


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

(1 + 119887120573)



+

(4)


120573) minus 120572(1 minus 119887)


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573

minus ℎ1198902

(5)






119868= 0



119868first If



119868ge 0

If 119873119868le 0 then 120582

lowast119890 = 119879 and (1 minus 120582



and 120582lowast119890 ge 119879 if 119873










119868= 0





119868le 0 and 119890lowast gt 119879 + 120591


119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

1 + 119887120573

+

(119901 minus 119888) [119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]

1 + 119887120573

minus ℎ(119879 + 120591)2

(6)



119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198681(119886)= 119890lowast

1198681(119886)= 119879 + 120591 120582

lowast

1198681(119886)=

119879

(119879 + 120591)

119901lowast

1198681(119886)=

119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591

2119866

+ 119888

(7)

120587119871119871(120596lowast

1198681(119886)) =

[119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]2

4119866 (1 + 119887120573)

minus ℎ(119879 + 120591)2

(8)






120582lowast1198681(119886)

and119901lowast


1198681(119886) 120582lowast1198681(119886)

and 119901lowast



119889119879 = 1 gt 0 119889120582lowast1198681(119886)

119889119879 =

120591(119879 + 120591)2gt 0 and119889119901lowast

1198681(119886)119889119879 = 119886(1+120573)(2119866) gt 0) In other



and 120582lowast1198681(119886)






1198681(119886)and

120587119871119871(120596lowast




119865(120596) gt 0

120582lowast= 1 minus 120591119890



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573

minus ℎ1198902

(9)


119890lowast

1198681(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) (10)

and 119890lowast1198681(119887)





119868ge 0 119863

119871(120596lowast

1198681(119887)) gt 0 and 119863

119865(120596lowast

1198681(119887)) gt 0



120596lowast

1198681(119887)= 119890lowast

1198681(119887) 120582lowast

1198681(119887) 119901lowast

1198681(119887) (11)

120587119871119871(120596lowast

1198681(119887)) =

ℎ(119861 minus 119873119868120591)2

119884

(12)

where 119890lowast1198681(119887)

= 119886(1 + 120573)(119861 minus 119873119868120591)119884 120582lowast

1198681(119887)= 1 minus 120591119890

lowast

1198681(119887)

and 119901lowast119868119894(119887)

= 119888 + 2ℎ(119861 minus 119873119868120591)(1 + 119887120573)119884



1198681(119887)gt 119888 if and only

if 119861 gt 119873119868120591 Because 119861 gt 119884(119879 + 120591)[119886(1 + 120573)] + 119873

119868120591 implies




120582lowast1198681(119886)

119901lowast1198681(119886)

and120587119871119871(120596lowast


lowast

1198681(119886)119889119879 = minus119884119861119886(1 +



1198681(119886) 120582lowast1198681(119886)

119901lowast

1198681(119886) and 120587

119871119871(120596lowast

1198681(119887))


1198681(119887)and 120587

119871119871(120596lowast




119865(120596) gt 0





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573

minus ℎ1198902

(13)





119890lowast

1198681(119888)(119901) =

120572 (1 minus 119887)

2ℎ (1 + 119887120573)

(119901 minus 119888) (14)

and 119890lowast1198681(119888)


only if (i) 119873119868le 0 119863

119871(120596lowast

1198681(119888)) gt 0 and 119863

119865(120596lowast

1198681(119888)) gt 0 (ii)



120596lowast

1198681(119888)= 119890lowast

1198681(119888) 120582lowast

1198681(119888) 119901lowast

1198681(119888) (15)

120587119871119871(120596lowast

1198681(119888)) =

ℎ(119861 + 119873119868119879)2

119885

(16)

where 119890lowast1198681(119888)

= 120572(1minus119887)(119861+119873119868119879)119885 120582lowast

1198681(119888)= 119879119885120572(1minus119887)(119861+

119873119868119879) and 119901lowast

1198681(119888)= 2ℎ(119861 + 119873

119868119879)(1 + 119887120573)119885 + 119888




119868119894(119888)gt 119888 if




1198681(119888)) 119890lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888) 120582lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888)and 120587

119871119871(120596lowast




120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(17)


119868(119901 minus 119888) lt 120583(1 + 119887120573) and


119868(119901 minus 119888) lt





119868(119901minus119888) gt 120583(1+119887120573)







119868(119901 minus 119888) = 120583(1 + 119887120573) then the







and


and 120582lowast = 119879119890


119871(120596) gt 0

and119863119865(120596) gt 0

Denote by 120596lowast1198682(119894)



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119879]

1 + 119887120573

minus ℎ1198792minus 120583120591

(18)



119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 + 119886(1 + 120573)119879 gt 0


120596lowast

1198682(119886)= 119890lowast

1198682(119886)= 119879 120582

lowast

1198682(119886)= 1

119901lowast

1198682(119886)= 119888 +

[119861 + 119886 (1 + 120573) 119879]

(2119866)

(19)

120587119871119871(120596lowast

1198682(119886)) =

1198612+ 2119861119886 (1 + 120573) 119879 minus 119884119879

2

4119866 (1 + 119887120573)

minus 120583120591 (20)



1198682(119886)gt 119888 Moreover


1198682(119886)and

119901lowast



1198682(119886)and 119901lowast

1198682(119886)are



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 120583120591

(21)


119873119868gt 0


119890lowast

1198682(119887)(119901) = 119890

lowast

1198681(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(22)

and 119890lowast1198682(119887)


only if (i) 119873119868gt 0 119863

119871(120596lowast

1198682(119887)) gt 0 and 119863

119865(120596lowast

1198682(119887)) gt 0 (ii)



120596lowast

1198682(119887)= 119890lowast

1198682(119887) 120582lowast

1198682(119887) 119901lowast

1198682(119887) (23)

120587119871119871(120596lowast

1198682(119887)) =

ℎ1198612

119884

minus 120583120591 (24)

where 119890lowast1198682(119887)

= 119886(1 + 120573)119861119884 120582lowast1198682(119887)

= 1 and 119901lowast1198682(119887)

= 2ℎ119861(1 +

119887120573)119884 + 119888





119868(119901 minus 119888) gt 120583(1 + 119887120573) which is




119871119871(120596lowast



1198682(119887) 120582lowast1198682(119887)

and 119901

lowast



1198682(119887) 120582lowast1198682(119887)

and 119901lowast1198682(119887)



119865(120596) gt 0


119868(119901minus 119888) = 120583(1+119887120573) implies

119873119868gt 0


119890lowast

1198682(119888)(119901) = 119890

lowast

1198682(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(25)

and 119890lowast1198682(119888)


only if (i)119873119868gt 0119863

119871(120596lowast

1198682(119888)) gt 0 and119863

119865(120596lowast

1198682(119888)) gt 0 and (ii)



120596lowast

1198682(119888)= 119890lowast

1198682(119888) 120582lowast

1198682(119888) 119901lowast

1198682(119888) (26)

120587119871119871(120596lowast

1198682(119888)) =

119861120583

119873119868

minus

1198841205832

4ℎ1198732

119868

minus 120583120591 (27)

where 119890lowast1198682(119888)

= 119886120583(1 + 120573)(2ℎ119873119868) 120582lowast1198682(119888)

= 1 and 119901lowast1198682(119888)

= 119888 +

120583(1 + 119887120573)119873119868







1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)


1198682(119888)and119901lowast

1198682(119888)are




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573


(28)


119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)


119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873



120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]







1198682(119889)gt 119888 Finally the




and 119901lowast

1198682(119889) we


and 119901lowast1198682(119889)









1198682(119889) 120582lowast1198682(119889)

and 119901lowast



120582lowast1198682(119889)

and 119901lowast1198682(119889)



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(32)



119868(119901 minus 119888) gt minus119898(1 + 119887120573)







119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)









119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119863 (120596) =


(1 + 119887120573)

(3)


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

(1 + 119887120573)



+

(4)


120573) minus 120572(1 minus 119887)


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573

minus ℎ1198902

(5)






119868= 0



119868first If



119868ge 0

If 119873119868le 0 then 120582

lowast119890 = 119879 and (1 minus 120582



and 120582lowast119890 ge 119879 if 119873










119868= 0





119868le 0 and 119890lowast gt 119879 + 120591


119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

1 + 119887120573

+

(119901 minus 119888) [119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]

1 + 119887120573

minus ℎ(119879 + 120591)2

(6)



119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198681(119886)= 119890lowast

1198681(119886)= 119879 + 120591 120582

lowast

1198681(119886)=

119879

(119879 + 120591)

119901lowast

1198681(119886)=

119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591

2119866

+ 119888

(7)

120587119871119871(120596lowast

1198681(119886)) =

[119861 + 119886 (1 + 120573) 119879 + 120572 (1 minus 119887) 120591]2

4119866 (1 + 119887120573)

minus ℎ(119879 + 120591)2

(8)






120582lowast1198681(119886)

and119901lowast


1198681(119886) 120582lowast1198681(119886)

and 119901lowast



119889119879 = 1 gt 0 119889120582lowast1198681(119886)

119889119879 =

120591(119879 + 120591)2gt 0 and119889119901lowast

1198681(119886)119889119879 = 119886(1+120573)(2119866) gt 0) In other



and 120582lowast1198681(119886)






1198681(119886)and

120587119871119871(120596lowast




119865(120596) gt 0

120582lowast= 1 minus 120591119890



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573

minus ℎ1198902

(9)


119890lowast

1198681(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) (10)

and 119890lowast1198681(119887)





119868ge 0 119863

119871(120596lowast

1198681(119887)) gt 0 and 119863

119865(120596lowast

1198681(119887)) gt 0



120596lowast

1198681(119887)= 119890lowast

1198681(119887) 120582lowast

1198681(119887) 119901lowast

1198681(119887) (11)

120587119871119871(120596lowast

1198681(119887)) =

ℎ(119861 minus 119873119868120591)2

119884

(12)

where 119890lowast1198681(119887)

= 119886(1 + 120573)(119861 minus 119873119868120591)119884 120582lowast

1198681(119887)= 1 minus 120591119890

lowast

1198681(119887)

and 119901lowast119868119894(119887)

= 119888 + 2ℎ(119861 minus 119873119868120591)(1 + 119887120573)119884



1198681(119887)gt 119888 if and only

if 119861 gt 119873119868120591 Because 119861 gt 119884(119879 + 120591)[119886(1 + 120573)] + 119873

119868120591 implies




120582lowast1198681(119886)

119901lowast1198681(119886)

and120587119871119871(120596lowast


lowast

1198681(119886)119889119879 = minus119884119861119886(1 +



1198681(119886) 120582lowast1198681(119886)

119901lowast

1198681(119886) and 120587

119871119871(120596lowast

1198681(119887))


1198681(119887)and 120587

119871119871(120596lowast




119865(120596) gt 0





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573

minus ℎ1198902

(13)





119890lowast

1198681(119888)(119901) =

120572 (1 minus 119887)

2ℎ (1 + 119887120573)

(119901 minus 119888) (14)

and 119890lowast1198681(119888)


only if (i) 119873119868le 0 119863

119871(120596lowast

1198681(119888)) gt 0 and 119863

119865(120596lowast

1198681(119888)) gt 0 (ii)



120596lowast

1198681(119888)= 119890lowast

1198681(119888) 120582lowast

1198681(119888) 119901lowast

1198681(119888) (15)

120587119871119871(120596lowast

1198681(119888)) =

ℎ(119861 + 119873119868119879)2

119885

(16)

where 119890lowast1198681(119888)

= 120572(1minus119887)(119861+119873119868119879)119885 120582lowast

1198681(119888)= 119879119885120572(1minus119887)(119861+

119873119868119879) and 119901lowast

1198681(119888)= 2ℎ(119861 + 119873

119868119879)(1 + 119887120573)119885 + 119888




119868119894(119888)gt 119888 if




1198681(119888)) 119890lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888) 120582lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888)and 120587

119871119871(120596lowast




120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(17)


119868(119901 minus 119888) lt 120583(1 + 119887120573) and


119868(119901 minus 119888) lt





119868(119901minus119888) gt 120583(1+119887120573)







119868(119901 minus 119888) = 120583(1 + 119887120573) then the







and


and 120582lowast = 119879119890


119871(120596) gt 0

and119863119865(120596) gt 0

Denote by 120596lowast1198682(119894)



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119879]

1 + 119887120573

minus ℎ1198792minus 120583120591

(18)



119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 + 119886(1 + 120573)119879 gt 0


120596lowast

1198682(119886)= 119890lowast

1198682(119886)= 119879 120582

lowast

1198682(119886)= 1

119901lowast

1198682(119886)= 119888 +

[119861 + 119886 (1 + 120573) 119879]

(2119866)

(19)

120587119871119871(120596lowast

1198682(119886)) =

1198612+ 2119861119886 (1 + 120573) 119879 minus 119884119879

2

4119866 (1 + 119887120573)

minus 120583120591 (20)



1198682(119886)gt 119888 Moreover


1198682(119886)and

119901lowast



1198682(119886)and 119901lowast

1198682(119886)are



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 120583120591

(21)


119873119868gt 0


119890lowast

1198682(119887)(119901) = 119890

lowast

1198681(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(22)

and 119890lowast1198682(119887)


only if (i) 119873119868gt 0 119863

119871(120596lowast

1198682(119887)) gt 0 and 119863

119865(120596lowast

1198682(119887)) gt 0 (ii)



120596lowast

1198682(119887)= 119890lowast

1198682(119887) 120582lowast

1198682(119887) 119901lowast

1198682(119887) (23)

120587119871119871(120596lowast

1198682(119887)) =

ℎ1198612

119884

minus 120583120591 (24)

where 119890lowast1198682(119887)

= 119886(1 + 120573)119861119884 120582lowast1198682(119887)

= 1 and 119901lowast1198682(119887)

= 2ℎ119861(1 +

119887120573)119884 + 119888





119868(119901 minus 119888) gt 120583(1 + 119887120573) which is




119871119871(120596lowast



1198682(119887) 120582lowast1198682(119887)

and 119901

lowast



1198682(119887) 120582lowast1198682(119887)

and 119901lowast1198682(119887)



119865(120596) gt 0


119868(119901minus 119888) = 120583(1+119887120573) implies

119873119868gt 0


119890lowast

1198682(119888)(119901) = 119890

lowast

1198682(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(25)

and 119890lowast1198682(119888)


only if (i)119873119868gt 0119863

119871(120596lowast

1198682(119888)) gt 0 and119863

119865(120596lowast

1198682(119888)) gt 0 and (ii)



120596lowast

1198682(119888)= 119890lowast

1198682(119888) 120582lowast

1198682(119888) 119901lowast

1198682(119888) (26)

120587119871119871(120596lowast

1198682(119888)) =

119861120583

119873119868

minus

1198841205832

4ℎ1198732

119868

minus 120583120591 (27)

where 119890lowast1198682(119888)

= 119886120583(1 + 120573)(2ℎ119873119868) 120582lowast1198682(119888)

= 1 and 119901lowast1198682(119888)

= 119888 +

120583(1 + 119887120573)119873119868







1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)


1198682(119888)and119901lowast

1198682(119888)are




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573


(28)


119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)


119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873



120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]







1198682(119889)gt 119888 Finally the




and 119901lowast

1198682(119889) we


and 119901lowast1198682(119889)









1198682(119889) 120582lowast1198682(119889)

and 119901lowast



120582lowast1198682(119889)

and 119901lowast1198682(119889)



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(32)



119868(119901 minus 119888) gt minus119898(1 + 119887120573)







119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)









119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












120582lowast1198681(119886)

and119901lowast


1198681(119886) 120582lowast1198681(119886)

and 119901lowast



119889119879 = 1 gt 0 119889120582lowast1198681(119886)

119889119879 =

120591(119879 + 120591)2gt 0 and119889119901lowast

1198681(119886)119889119879 = 119886(1+120573)(2119866) gt 0) In other



and 120582lowast1198681(119886)






1198681(119886)and

120587119871119871(120596lowast




119865(120596) gt 0

120582lowast= 1 minus 120591119890



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573

minus ℎ1198902

(9)


119890lowast

1198681(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) (10)

and 119890lowast1198681(119887)





119868ge 0 119863

119871(120596lowast

1198681(119887)) gt 0 and 119863

119865(120596lowast

1198681(119887)) gt 0



120596lowast

1198681(119887)= 119890lowast

1198681(119887) 120582lowast

1198681(119887) 119901lowast

1198681(119887) (11)

120587119871119871(120596lowast

1198681(119887)) =

ℎ(119861 minus 119873119868120591)2

119884

(12)

where 119890lowast1198681(119887)

= 119886(1 + 120573)(119861 minus 119873119868120591)119884 120582lowast

1198681(119887)= 1 minus 120591119890

lowast

1198681(119887)

and 119901lowast119868119894(119887)

= 119888 + 2ℎ(119861 minus 119873119868120591)(1 + 119887120573)119884



1198681(119887)gt 119888 if and only

if 119861 gt 119873119868120591 Because 119861 gt 119884(119879 + 120591)[119886(1 + 120573)] + 119873

119868120591 implies




120582lowast1198681(119886)

119901lowast1198681(119886)

and120587119871119871(120596lowast


lowast

1198681(119886)119889119879 = minus119884119861119886(1 +



1198681(119886) 120582lowast1198681(119886)

119901lowast

1198681(119886) and 120587

119871119871(120596lowast

1198681(119887))


1198681(119887)and 120587

119871119871(120596lowast




119865(120596) gt 0





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573

minus ℎ1198902

(13)





119890lowast

1198681(119888)(119901) =

120572 (1 minus 119887)

2ℎ (1 + 119887120573)

(119901 minus 119888) (14)

and 119890lowast1198681(119888)


only if (i) 119873119868le 0 119863

119871(120596lowast

1198681(119888)) gt 0 and 119863

119865(120596lowast

1198681(119888)) gt 0 (ii)



120596lowast

1198681(119888)= 119890lowast

1198681(119888) 120582lowast

1198681(119888) 119901lowast

1198681(119888) (15)

120587119871119871(120596lowast

1198681(119888)) =

ℎ(119861 + 119873119868119879)2

119885

(16)

where 119890lowast1198681(119888)

= 120572(1minus119887)(119861+119873119868119879)119885 120582lowast

1198681(119888)= 119879119885120572(1minus119887)(119861+

119873119868119879) and 119901lowast

1198681(119888)= 2ℎ(119861 + 119873

119868119879)(1 + 119887120573)119885 + 119888




119868119894(119888)gt 119888 if




1198681(119888)) 119890lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888) 120582lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888)and 120587

119871119871(120596lowast




120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(17)


119868(119901 minus 119888) lt 120583(1 + 119887120573) and


119868(119901 minus 119888) lt





119868(119901minus119888) gt 120583(1+119887120573)







119868(119901 minus 119888) = 120583(1 + 119887120573) then the







and


and 120582lowast = 119879119890


119871(120596) gt 0

and119863119865(120596) gt 0

Denote by 120596lowast1198682(119894)



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119879]

1 + 119887120573

minus ℎ1198792minus 120583120591

(18)



119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 + 119886(1 + 120573)119879 gt 0


120596lowast

1198682(119886)= 119890lowast

1198682(119886)= 119879 120582

lowast

1198682(119886)= 1

119901lowast

1198682(119886)= 119888 +

[119861 + 119886 (1 + 120573) 119879]

(2119866)

(19)

120587119871119871(120596lowast

1198682(119886)) =

1198612+ 2119861119886 (1 + 120573) 119879 minus 119884119879

2

4119866 (1 + 119887120573)

minus 120583120591 (20)



1198682(119886)gt 119888 Moreover


1198682(119886)and

119901lowast



1198682(119886)and 119901lowast

1198682(119886)are



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 120583120591

(21)


119873119868gt 0


119890lowast

1198682(119887)(119901) = 119890

lowast

1198681(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(22)

and 119890lowast1198682(119887)


only if (i) 119873119868gt 0 119863

119871(120596lowast

1198682(119887)) gt 0 and 119863

119865(120596lowast

1198682(119887)) gt 0 (ii)



120596lowast

1198682(119887)= 119890lowast

1198682(119887) 120582lowast

1198682(119887) 119901lowast

1198682(119887) (23)

120587119871119871(120596lowast

1198682(119887)) =

ℎ1198612

119884

minus 120583120591 (24)

where 119890lowast1198682(119887)

= 119886(1 + 120573)119861119884 120582lowast1198682(119887)

= 1 and 119901lowast1198682(119887)

= 2ℎ119861(1 +

119887120573)119884 + 119888





119868(119901 minus 119888) gt 120583(1 + 119887120573) which is




119871119871(120596lowast



1198682(119887) 120582lowast1198682(119887)

and 119901

lowast



1198682(119887) 120582lowast1198682(119887)

and 119901lowast1198682(119887)



119865(120596) gt 0


119868(119901minus 119888) = 120583(1+119887120573) implies

119873119868gt 0


119890lowast

1198682(119888)(119901) = 119890

lowast

1198682(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(25)

and 119890lowast1198682(119888)


only if (i)119873119868gt 0119863

119871(120596lowast

1198682(119888)) gt 0 and119863

119865(120596lowast

1198682(119888)) gt 0 and (ii)



120596lowast

1198682(119888)= 119890lowast

1198682(119888) 120582lowast

1198682(119888) 119901lowast

1198682(119888) (26)

120587119871119871(120596lowast

1198682(119888)) =

119861120583

119873119868

minus

1198841205832

4ℎ1198732

119868

minus 120583120591 (27)

where 119890lowast1198682(119888)

= 119886120583(1 + 120573)(2ℎ119873119868) 120582lowast1198682(119888)

= 1 and 119901lowast1198682(119888)

= 119888 +

120583(1 + 119887120573)119873119868







1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)


1198682(119888)and119901lowast

1198682(119888)are




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573


(28)


119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)


119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873



120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]







1198682(119889)gt 119888 Finally the




and 119901lowast

1198682(119889) we


and 119901lowast1198682(119889)









1198682(119889) 120582lowast1198682(119889)

and 119901lowast



120582lowast1198682(119889)

and 119901lowast1198682(119889)



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(32)



119868(119901 minus 119888) gt minus119898(1 + 119887120573)







119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)









119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119890lowast

1198681(119888)(119901) =

120572 (1 minus 119887)

2ℎ (1 + 119887120573)

(119901 minus 119888) (14)

and 119890lowast1198681(119888)


only if (i) 119873119868le 0 119863

119871(120596lowast

1198681(119888)) gt 0 and 119863

119865(120596lowast

1198681(119888)) gt 0 (ii)



120596lowast

1198681(119888)= 119890lowast

1198681(119888) 120582lowast

1198681(119888) 119901lowast

1198681(119888) (15)

120587119871119871(120596lowast

1198681(119888)) =

ℎ(119861 + 119873119868119879)2

119885

(16)

where 119890lowast1198681(119888)

= 120572(1minus119887)(119861+119873119868119879)119885 120582lowast

1198681(119888)= 119879119885120572(1minus119887)(119861+

119873119868119879) and 119901lowast

1198681(119888)= 2ℎ(119861 + 119873

119868119879)(1 + 119887120573)119885 + 119888




119868119894(119888)gt 119888 if




1198681(119888)) 119890lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888) 120582lowast1198681(119888)

and 119901lowast1198681(119888)



1198681(119888)and 120587

119871119871(120596lowast




120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(17)


119868(119901 minus 119888) lt 120583(1 + 119887120573) and


119868(119901 minus 119888) lt





119868(119901minus119888) gt 120583(1+119887120573)







119868(119901 minus 119888) = 120583(1 + 119887120573) then the







and


and 120582lowast = 119879119890


119871(120596) gt 0

and119863119865(120596) gt 0

Denote by 120596lowast1198682(119894)



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119879]

1 + 119887120573

minus ℎ1198792minus 120583120591

(18)



119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 + 119886(1 + 120573)119879 gt 0


120596lowast

1198682(119886)= 119890lowast

1198682(119886)= 119879 120582

lowast

1198682(119886)= 1

119901lowast

1198682(119886)= 119888 +

[119861 + 119886 (1 + 120573) 119879]

(2119866)

(19)

120587119871119871(120596lowast

1198682(119886)) =

1198612+ 2119861119886 (1 + 120573) 119879 minus 119884119879

2

4119866 (1 + 119887120573)

minus 120583120591 (20)



1198682(119886)gt 119888 Moreover


1198682(119886)and

119901lowast



1198682(119886)and 119901lowast

1198682(119886)are



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 120583120591

(21)


119873119868gt 0


119890lowast

1198682(119887)(119901) = 119890

lowast

1198681(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(22)

and 119890lowast1198682(119887)


only if (i) 119873119868gt 0 119863

119871(120596lowast

1198682(119887)) gt 0 and 119863

119865(120596lowast

1198682(119887)) gt 0 (ii)



120596lowast

1198682(119887)= 119890lowast

1198682(119887) 120582lowast

1198682(119887) 119901lowast

1198682(119887) (23)

120587119871119871(120596lowast

1198682(119887)) =

ℎ1198612

119884

minus 120583120591 (24)

where 119890lowast1198682(119887)

= 119886(1 + 120573)119861119884 120582lowast1198682(119887)

= 1 and 119901lowast1198682(119887)

= 2ℎ119861(1 +

119887120573)119884 + 119888





119868(119901 minus 119888) gt 120583(1 + 119887120573) which is




119871119871(120596lowast



1198682(119887) 120582lowast1198682(119887)

and 119901

lowast



1198682(119887) 120582lowast1198682(119887)

and 119901lowast1198682(119887)



119865(120596) gt 0


119868(119901minus 119888) = 120583(1+119887120573) implies

119873119868gt 0


119890lowast

1198682(119888)(119901) = 119890

lowast

1198682(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(25)

and 119890lowast1198682(119888)


only if (i)119873119868gt 0119863

119871(120596lowast

1198682(119888)) gt 0 and119863

119865(120596lowast

1198682(119888)) gt 0 and (ii)



120596lowast

1198682(119888)= 119890lowast

1198682(119888) 120582lowast

1198682(119888) 119901lowast

1198682(119888) (26)

120587119871119871(120596lowast

1198682(119888)) =

119861120583

119873119868

minus

1198841205832

4ℎ1198732

119868

minus 120583120591 (27)

where 119890lowast1198682(119888)

= 119886120583(1 + 120573)(2ℎ119873119868) 120582lowast1198682(119888)

= 1 and 119901lowast1198682(119888)

= 119888 +

120583(1 + 119887120573)119873119868







1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)


1198682(119888)and119901lowast

1198682(119888)are




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573


(28)


119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)


119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873



120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]







1198682(119889)gt 119888 Finally the




and 119901lowast

1198682(119889) we


and 119901lowast1198682(119889)









1198682(119889) 120582lowast1198682(119889)

and 119901lowast



120582lowast1198682(119889)

and 119901lowast1198682(119889)



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(32)



119868(119901 minus 119888) gt minus119898(1 + 119887120573)







119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)









119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast

1198682(119886)) gt 0 (ii) 119866 gt 0 and (iii) 119861 + 119886(1 + 120573)119879 gt 0


120596lowast

1198682(119886)= 119890lowast

1198682(119886)= 119879 120582

lowast

1198682(119886)= 1

119901lowast

1198682(119886)= 119888 +

[119861 + 119886 (1 + 120573) 119879]

(2119866)

(19)

120587119871119871(120596lowast

1198682(119886)) =

1198612+ 2119861119886 (1 + 120573) 119879 minus 119884119879

2

4119866 (1 + 119887120573)

minus 120583120591 (20)



1198682(119886)gt 119888 Moreover


1198682(119886)and

119901lowast



1198682(119886)and 119901lowast

1198682(119886)are



119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573

minus ℎ1198902minus 120583120591

(21)


119873119868gt 0


119890lowast

1198682(119887)(119901) = 119890

lowast

1198681(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(22)

and 119890lowast1198682(119887)


only if (i) 119873119868gt 0 119863

119871(120596lowast

1198682(119887)) gt 0 and 119863

119865(120596lowast

1198682(119887)) gt 0 (ii)



120596lowast

1198682(119887)= 119890lowast

1198682(119887) 120582lowast

1198682(119887) 119901lowast

1198682(119887) (23)

120587119871119871(120596lowast

1198682(119887)) =

ℎ1198612

119884

minus 120583120591 (24)

where 119890lowast1198682(119887)

= 119886(1 + 120573)119861119884 120582lowast1198682(119887)

= 1 and 119901lowast1198682(119887)

= 2ℎ119861(1 +

119887120573)119884 + 119888





119868(119901 minus 119888) gt 120583(1 + 119887120573) which is




119871119871(120596lowast



1198682(119887) 120582lowast1198682(119887)

and 119901

lowast



1198682(119887) 120582lowast1198682(119887)

and 119901lowast1198682(119887)



119865(120596) gt 0


119868(119901minus 119888) = 120583(1+119887120573) implies

119873119868gt 0


119890lowast

1198682(119888)(119901) = 119890

lowast

1198682(119887)(119901) =

119886 (1 + 120573) (119901 minus 119888)

[2ℎ (1 + 119887120573)]

(25)

and 119890lowast1198682(119888)


only if (i)119873119868gt 0119863

119871(120596lowast

1198682(119888)) gt 0 and119863

119865(120596lowast

1198682(119888)) gt 0 and (ii)



120596lowast

1198682(119888)= 119890lowast

1198682(119888) 120582lowast

1198682(119888) 119901lowast

1198682(119888) (26)

120587119871119871(120596lowast

1198682(119888)) =

119861120583

119873119868

minus

1198841205832

4ℎ1198732

119868

minus 120583120591 (27)

where 119890lowast1198682(119888)

= 119886120583(1 + 120573)(2ℎ119873119868) 120582lowast1198682(119888)

= 1 and 119901lowast1198682(119888)

= 119888 +

120583(1 + 119887120573)119873119868







1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)


1198682(119888)and119901lowast

1198682(119888)are




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573


(28)


119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)


119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873



120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]







1198682(119889)gt 119888 Finally the




and 119901lowast

1198682(119889) we


and 119901lowast1198682(119889)









1198682(119889) 120582lowast1198682(119889)

and 119901lowast



120582lowast1198682(119889)

and 119901lowast1198682(119889)



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(32)



119868(119901 minus 119888) gt minus119898(1 + 119887120573)







119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)









119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1198682(119888) 120582lowast1198682(119888)

and 119901lowast1198682(119888)


1198682(119888)and119901lowast

1198682(119888)are




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 119879119873119868]

1 + 119887120573


(28)


119890lowast

1198682(119889)(119901) =

120572 (119901 minus 119888) (1 minus 119887)

[2ℎ (1 + 119887120573)]

+ 120583 (29)


119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 (ii) 119885 gt 0 (iii)

119879 lt 119890lowast

1198682(119889)lt 119879 + 120591 (iv) 2ℎ(119861 + 119873

119868119879) + 120572120583(1 minus 119887) gt 0 and (v)

119873119868[2ℎ(119861 + 119873



120596lowast

1198682(119889)= 119890lowast

1198682(119889) 120582lowast

1198682(119889) 119901lowast

1198682(119889) (30)

120587119871119871(120596lowast

1198682(119889)) =

[2ℎ119861 + 2ℎ119873119868119879 + 120572120583 (1 minus 119887)]

2

+ 1205832119885

4ℎ119885

minus 120583 (119879 + 120591)

(31)

where 119890lowast1198682(119889)

= 120572(1minus119887)[2ℎ(119861+119873119868119879)+120572120583(1minus119887)]2ℎ119885+1205832ℎ

120582lowast

1198682(119889)= 119879119890

lowast

1198682(119889) and 119901lowast

1198682(119889)= (1+119887120573)[2ℎ(119861+119873

119868119879)+120572120583(1minus

119887)]119885 + 119888 120587119871119871(120596) = (minus119866(119901 minus 119888)

2+ (119901 minus 119888)[119861 + 120572(1 minus 119887)119890 +

119879119873119868])(1 + 119887120573) minus ℎ120583

2minus 120583[119879 + 120591 minus 120583]







1198682(119889)gt 119888 Finally the




and 119901lowast

1198682(119889) we


and 119901lowast1198682(119889)









1198682(119889) 120582lowast1198682(119889)

and 119901lowast



120582lowast1198682(119889)

and 119901lowast1198682(119889)



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890 + 120582119873119868119890]

1 + 119887120573


(32)



119868(119901 minus 119888) gt minus119898(1 + 119887120573)







119868(119901minus 119888) +119898(1+ 119887120573) If119873

119868(119901minus 119888) gt minus119898(1+ 119887120573)









119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119868(119901 minus 119888) =


119868(119901 minus 119888) =





119868lt 0119873

119868(119901 minus 119888) lt minus119898(1 + 119887120573)


119868lt 0119873

119868(119901 minus 119888) = minus119898(1 + 119887120573)

119890 gt 120591 and 120582lowast = 0


119871(120596) gt 0

and119863119865(120596) gt 0




119865(120596) gt 0



120587119871119871(120596) =


1 + 119887120573

minus ℎ1198792minus 119898119879

(33)


119871(120596lowast

1198683(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198683(119886)= 119890lowast

1198683(119886)= 120591 120582

lowast

1198683(119886)= 0

119901lowast

1198683(119886)= 119888 +

[119861 + 120572 (1 minus 119887120573) 120591]

(2119866)

(34)

120587119871119871(120596lowast

1198683(119886)) =

1198612+ 2119861120572 (1 minus 119887) 120591 minus 119885120591

2

4119866 (1 + 119887120573)

minus 119898119879 (35)



1198683(119886)gt 119888 Moreover




and 119901lowast1198683(119886)



119901lowast


119901lowast




119865(120596) gt 0



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890 minus 120591119873119868]

1 + 119887120573


(36)


119890lowast

1198683(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

(37)

Moreover 119890lowast1198683(119887)


only if (i) 119863119871(120596lowast

1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 (ii) 119884 gt 0 (iii)




and 119884 = 4ℎ119866(1 + 119887120573) minus 1198862(1 + 120573)


120596lowast

1198683(119887)= 119890lowast

1198683(119887) 120582lowast

1198683(119887) 119901lowast

1198683(119887) (38)

120587119871119871(120596lowast

1198683(119887)) =

[2ℎ (119861 minus 119873119868120591) + 119886119898 (1 + 120573)]

2

4ℎ2119884

minus 119898 (119879 + 120591) +

1198982

4ℎ

(39)

where 119890lowast1198683(119887)

= 119886(1+120573)[2ℎ(119861minus119873119868120591)+119886119898(1+120573)]2ℎ119884+1198982ℎ

120582lowast

1198683(119887)= 1 minus 120591119890

lowast

1198683(119887) and 119901lowast

1198683(119887)= 119888 + (1 + 119887120573)[2ℎ(119861 minus119873

119868120591) +

119886119898(1 + 120573)]119884




119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119868(119901 minus 119888) gt



120582lowast

1198683(119887) and 119901

lowast


1198683(119887) 120582lowast1198683(119887)

and119901lowast

1198683(119887)depend on 119873







if119873119868= 0


119865(120596) gt 0

119873119868lt 0119873



120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573

minus ℎ1198902minus 119898119879

(40)


119890lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(41)

and 119890lowast1198683(119888)


only if (i)119873119868lt 0 119863

119871(120596lowast

1198683(119888)) gt 0 and 119863

119865(120596lowast

1198683(119888)) gt 0 (ii) 119885 gt


119885 = 4ℎ119866(1 + 119887120573) minus 1205722(1 minus 119887)


120596lowast

1198683(119888)= 119890lowast

1198683(119888)=

120572119861 (1 minus 119887)

119885

120582lowast

1198683(119888)= 0

119901lowast

1198683(119888)= 119888 +

2ℎ119861 (1 + 119887120573)

119885

(42)

120587119871119871(120596lowast

1198683(119888)) =

ℎ1198612

119885

minus 119898119879 (43)


1198683(119888) Plus


120582lowast1198683(119888)

and 119901lowast1198683(119888)



119865(120596) gt 0

119887 lt 1119873119868lt 0119873



119890lowast

1198683(119889)(119901) = 119890

lowast

1198683(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

(44)

and 119890lowast1198683(119888)


only if (i) 119873119868lt 0 119863

119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast

1198683(119889)) gt 0 (ii)



120596lowast

1198683(119889)= 119890lowast

1198683(119889)=

120572119898 (119887 minus 1)

(2ℎ119873119868)

120582lowast

1198683(119889)= 0

119901lowast

1198683(119889)= 119888 minus

119898 (1 + 119887120573)

119873119868

(45)

120587119871119871(120596lowast

1198683(119889)) =

12057221198982(1 minus 119887)

2minus 4ℎ119898 [119866119898 (1 + 119887120573) + 119861119873

119868]

4ℎ1198732

119868

minus 119898119879

(46)




1198683(119889) 120582lowast1198683(119889)

and 119901lowast1198683(119889)


1198683(119889)and 119901

lowast

1198683(119889)are

increasing in119898


119865(120596) gt 0 120582119890 lt


120587119871119871(120596) =


2

1 + 119887120573


(47)






















119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























119865(120596) gt 0






120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) 119861

1 + 119887120573



119871(120596lowast

1198684(119886)) gt 0 and

119863119865(120596lowast



120596lowast

1198684(119886)= 119890lowast

1198684(119886)= 0 120582

lowast

1198684(119886)= 0 119901

lowast

1198684(119886)=

119861

(2G)+ c (49)

120587119871119871(120596lowast

1198684(119886)) =

1198612

4119866 (1 + 119887120573)

minus 119898119879 minus 120583120591 (50)




0 lt 119890 lt 119879 and 120582lowast = 1


120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 119886 (1 + 120573) 119890]

1 + 119887120573


(51)


119890lowast

1198684(119887)(119901) =

119886 (1 + 120573)

2ℎ (1 + 119887120573)

(119901 minus 119888) +

119898

2ℎ

= 119890lowast

1198683(119887)(119901) (52)

Moreover 119890lowast1198684(119887)


only if (i) 119863119871(120596lowast

1198684(119887)) gt 0 and 119863

119865(120596lowast

1198684(119887)) gt 0 (ii) 119884 gt 0 (iii)


119868119861 gt


120596lowast

1198684(119887)= 119890lowast

1198684(119887) 120582lowast

1198684(119887) 119901lowast

1198684(119887) (53)

120587119871119871(120596lowast

1198684(119887)) =

ℎ1198612minus 119861119886119898 (1 + 120573) + 119898

2119866 (1 + 119887120573)

119884

minus 119898119879 minus 120583120591

(54)

where 119890lowast1198684(119887)

= (119886(1 + 120573)119861 + 2119898119866(1 + 119887120573))119884 120582lowast1198684(119887)

= 1 and119901lowast

1198684(119887)= [2ℎ119861 + 119886119898(1 + 120573)](1 + 119887120573)119884 + 119888


1198684(119887)



1198684(119887) 120582lowast1198684(119887)

and 119901lowast1198684(119887)





120587119871119871(120596) =

minus119866(119901 minus 119888)2

+ (119901 minus 119888) [119861 + 120572 (1 minus 119887) 119890]

1 + 119887120573


(55)



119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119890lowast

1198684(119888)(119901) =

120572 (1 minus 119887) (119901 minus 119888)

2ℎ (1 + 119887120573)

+

120583

2ℎ

(56)

Note that 119890lowast1198684(119888)





119871(120596lowast

1198684(119888)) gt 0 and

119863119865(120596lowast




120596lowast

1198684(119888)= 119890lowast

1198684(119888) 120582lowast

1198684(119888) 119901lowast

1198684(119888)

120587119871119871(120596lowast

1198684(119888)) =

ℎ1198612

119885

minus 119898119879

(57)

where 119890lowast1198684(119888)

= [2120583119866(1 + 119887120573) + 120572(1 minus 119887)119861]119885 120582lowast1198684(119888)

= 0 and119901lowast

1198684(119888)= (1 + 119887120573)[2ℎ119861 + 120572120583(1 minus 119887)]119885 + 119888



119868(119901minus


120582lowast1198684(119888)

and 119901lowast1198684(119888)












Appendix

Proofs


119871119871(120596)120597120582 =


119890 gt 119879 + 120591 if 119873119868ge 0 then 120587





119868le 0


lowast= Min120582 120582119890 ge



119873119868= 0 120597120587

119871119871(120596)120597120582 = 0 so 120587




119871(120596lowast

1198681(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879 + 120572(1 minus 119887)120591])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2





1198681(119886)=

arg119901120597120587119871119871(120596)120597119901 = 0 = [119861 + 119886(1 + 120573)119879 + 120572(1 minus 119887)120591]2119866 + 119888


Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Since 119901lowast1198681(119886)


1198681(119886)into (6) we obtain (8)


119871119871(120596)120597119890 = 119886(119901 minus 119888)(1 + 120573)(1 + 119887120573) minus 2ℎ119890

and 1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave





119868ge 0 119863

119871(120596lowast

119868119894(119887)) gt

0 and 119863119865(120596lowast



minus119884(119901 minus 119888)24ℎ(1 + 119887120573)

2+ (119861 minus 120591119873

119868)(119901 minus 119888)(1 + 119887120573) and for


2+

(119861 minus 120591119873119868)(1 + 119887120573) and 120597

2120587119871119871(120596)120597119901

2= minus1198842ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119887)

gt 119888 we obtain 119861 gt 119873119868120591



1198681(119887) By con-

sidering 119890lowast1198681(119887)



120596lowast

1198681(119887)into (9) we obtain (12)


119871119871(120596)


119871119871(120596) is a con-


119887)[2ℎ(1 + 119887120573)] gt 0 119890lowast1198681(119888)



119868le 0119863

119871(120596lowast

119868119894(119888)) gt 0 and119863

119865(120596lowast

119868119894(119888)) gt 0 are basic


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ (119861 + 119879119873

119868)(119901 minus


2+

(119861 + 119879119873119868)(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587




Since 119901lowast1198681(119888)



1198681(119888)into (14) we obtain

119890lowast




putting 120596lowast1198681(119888)



119871119871(120596)120597120582 = 119873

119868119890(119901 minus 119888)(1 + 119887120573) minus 120583119890




120583(1 + 119887120573) If119873119868(119901 minus 119888) lt 120583(1 + 119887120573) then 120587

119871119871(120596) is decreasing



119868(119901 minus

119888) lt 120583(1 + 119887120573) 0 le 12058210158401015840lt 1205821015840le 1 119863

119871(1205961015840) gt 0 119863

119865(1205961015840) gt 0

119863119871(12059610158401015840) gt 0119863


(1minus12058210158401015840) = 120591 We have 120587



If 119873119868(119901 minus 119888) = 120583(1 + 119887120573) then 120587



119868(119901 minus 119888) = 120583(1 + 119887120573)


119871(120596lowast

1198682(119886)) gt 0 and

119863119865(120596lowast



119871119871(120596)120597119901 = (minus2119866(119901 minus 119888) + [119861 +

119886(1 + 120573)119879])(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus2119866 where 119866 is




1198682(119886)

Since 119901lowast1198682(119886)


1198682(119886)into (18) we obtain (20)


119871119871(120596)






119868gt 0119863

119871(120596lowast

1198682(119887)) gt

0 and 119863119865(120596lowast



24ℎ(1 +


119871119871(120596)120597119901 =

minus119884(119901minus119888)2ℎ(1 + 119887120573)2+119861(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2








119868(119901lowast





119871(120596lowast

1198682(119888)) gt 0 and 119863

119865(120596lowast



obtain 119901lowast1198682(119888)


1198682(119888)into


1198682(119888)gt 119879 we




1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



119871(120596lowast

1198682(119889)) gt 0 and 119863

119865(120596lowast

1198682(119889)) gt 0 are the basic


24ℎ(1 + 119887120573)

2+ [2ℎ(119861 + 119879119873

119868) + 120572120583(1 minus


119871119871(120596)

120597119901 = minus119885(119901minus119888)2ℎ(1 + 119887120573)2+(2ℎ(119861+119879119873

119868)+120572120583(1minus119887))2ℎ(1+

119887120573) and 1205972120587119871119871(120596)120597119901

2= minus1198852ℎ(1 + 119887120573)

2Therefore 120587





1198682(119889)lt







lowast


1198682(119889)into (28) we obtain

(31)


119871119871(120596)120597120582 = 119873

119868119890(119901minus 119888)(1+ 119887120573)+119898119890




1205961015840= (119890 120582

1015840 119901) and 12059610158401015840 = (119890 120582

10158401015840 119901) where 119890 ge 119879 + 120591 119901 gt 119888

119873119868(119901 minus 119888) gt minus119898(1 + 119887120573) 0 le 120582

1015840lt 12058210158401015840le 1 119863

119871(1205961015840) gt 0

119863119865(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(12059610158401015840) gt 0 1205821015840119890 lt 119879 12058210158401015840119890 = 119879


119871119871(1205961015840) lt 120587119871119871(12059610158401015840)





119873119868(119901 minus 119888) = minus119898(1 + 119887120573) then 120587



119868(119901 minus 119888) =

minus119898(1 + 119887120573)


119871(120596lowast

1198683(119886)) gt 0

and 119863119865(120596lowast



1205972120587119871119871(120596)120597119901

2




1198683(119886) As 119901 gt 119888 we obtain


1198683(119886)into

(33) we obtain (34)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is



1198683(119887)) gt 0 and 119863

119865(120596lowast

1198683(119887)) gt 0 are the


119871119871(120596) = minus119884(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ [2ℎ(119861 minus 119873

119868120591) +



2+ [2ℎ(119861 minus 119873

119868120591) +

119886119898(1 + 120573)]2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2

= minus1198842ℎ(1 + 119887120573)2






119890lowast

1198683(119887)lt 119879 + 120591 and 119873

119868(119901lowast





(36) we obtain (39)


119871119871(120596)


119871119871(120596) is


1198683(119888)(119901) is strictly



119871(120596lowast

1198683(119888)) gt 0 and119863

119865(120596lowast

1198683(119888)) gt


119871119871(120596) = minus119885(119901 minus 119888)

24ℎ(1 + 119887120573)

2+ 119861(119901 minus


119888)2ℎ(1 + 119887120573)2+ 119861(1 + 119887120573) and 1205972120587

119871119871(120596)120597119901

2= minus1198852ℎ(1 +

119887120573)2 Therefore 120587








1198683(119888)minus 119888) lt minus119898(1 + 119887120573)


1198683(119888)



119868lt 0

119863119871(120596lowast

1198683(119889)) gt 0 and 119863

119865(120596lowast





1198683(119889)gt 0 as 119873

119868lt 0 for


1198683(119889)= 119890lowast

1198683(119889)(119901lowast


1198683(119889)into (17) we obtain





(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(1198901015840 12058210158401015840 1199011015840) where 0 lt 120582

1015840lt 12058210158401015840lt 1 119879 le 119890

1015840lt 119879 + 120591119873

119868(1199011015840minus

119888) gt (120583 minus 119898)(1 + 119887120573) 119863119871(1205961015840) gt 0 119863

119871(12059610158401015840) gt 0 119863

119865(1205961015840) gt




119871119871(1205961015840) However








119868(119901 minus 119888) lt (120583 minus






119868(119901minus119888) =

(120583 minus 119898)(1 + 119887120573)


1198684(119886)) gt 0 and 119863

119865(120596lowast



2120587119871119871(120596)

1205971199012= minus2119866 Therefore 120587




119901lowast


1198684(119886)into (48) we

obtain (50)


1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave


1198683(119887)(119901) is





119871(120596lowast

1198684(119887)) gt 0 and

119863119865(120596lowast



119888)24ℎ(1 + 119887120573)


11989824ℎ and we have 120597120587

119871119871(120596)120597119901 = minus119884(119901 minus 119888)2ℎ(1 + 119887120573)

2+

(2ℎ119861 + 119886119898(1 + 120573))2ℎ(1 + 119887120573) and 1205972120587119871119871(120596)120597119901

2= minus119884

2ℎ(1 + 119887120573)2






1198684(119887)into

(52) we obtain 119890lowast1198684(119887)


1198684(119887)minus 119888) gt (120583 minus

119898)(1 + 119887120573) and 0 lt 119890lowast





1205972120587119871119871(120596)120597119890


119871119871(120596) is a concave



1198684(119888)) gt 0 and 119863

119865(120596lowast



minus119885(119901 minus 119888)24ℎ(1 + 119887120573)



119871119871(120596)120597119901 = minus119885(119901 minus 119888)

2ℎ(1 + 119887120573)2+(2ℎ119861+120572120583(1minus119887))2ℎ(1+119887120573) and 1205972120587

119871119871(120596)120597119901

2=

minus1198852ℎ(1 + 119887120573)2






1198684(119888)lt

120591 and 119873119868(119901lowast



1198684(119888)into (55) we obtain (56)

Acknowledgement


References












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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