Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 127615 7 pageshttpdxdoiorg1011552013127615

Research ArticleFekete-Szegy Inequality for a Subclass of119901-Valent Analytic Functions

Mohsan Raza1 Muhammad Arif2 and Maslina Darus3

1 Department of Mathematics Government College University Faisalabad Faisalabad Punjab 38000 Pakistan2Department of Mathematics Abdul Wali Khan University Mardan Mardan Khyber Pakhtunkhwa 23200 Pakistan3 School of Mathematical Sciences Faculty of Science and Technology Universiti Kebangsaan Malaysia 43600 BangiSelangor D Ehsan Malaysia

Correspondence should be addressed to Muhammad Arif marifmathsyahoocom

Received 18 January 2013 Accepted 11 April 2013

Academic Editor Mina Abd-El-Malek

Copyright copy 2013 Mohsan Raza 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

Themain object of this paper is to study Fekete-Szego problem for the class of 119901-valent functions Fekete-Szego inequality of severalclasses is obtained as special cases from our results Applications of the results are also obtained on the class defined by convolution

1 Introduction and Preliminaries

Let 119860119901 denote the class of functions 119891(119911) of the form

119891 (119911) = 119911119901+

infin

sum

119899=119901+1

119886119899119911119899 (119901 isin N = 1 2 3 ) (1)

which are analytic in the open unit disk 119864 Also 1198601 = 119860 theusual class of analytic functions defined in the open unit disk119864 = 119911 |119911| lt 1 Let 119891(119911) and 119892(119911) be analytic in 119864 We saythat the function 119891 is subordinate to the function 119892 and write119891(119911) ≺ 119892(119911) if and only if there exists Schwarz function 119908analytic in 119864 such that 119908(0) = 0 |119908(119911)| lt 1 for 119911 isin 119864 and119891(119911) = 119892(119908(119911)) In particular if 119892 is univalent in 119864 then wehave the following equivalence

119891 (119911) ≺ 119892 (119911) lArrrArr 119891 (0) = 119892 (0) 119891 (119864) sub 119892 (119864) (2)

For any two analytic functions 119891(119911) of the form (1) and119892(119911) with

119892 (119911) = 119911119901+

infin

sum

119899=119901+1

119887119899119911119899 119911 isin 119864 (3)

the convolution (Hadamard product) is given by

(119891 lowast 119892) (119911) = 119911119901+

infin

sum

119899=119901+1

119886119899119887119899119911119899 119911 isin 119864 (4)

Let 120601(119911) be an analytic function with positive real part on 119864with 120601(0) = 1 1206011015840(0) gt 0 which maps the unit disk 119864 ontoa region starlike with respect to 1 which is symmetric withrespect to the real axis Denote by 119878lowast

119901(120601) the class of functions

119891 analytic in 119864 for which

1199111198911015840(119911)

119901119891 (119911)≺ 120601 (119911) 119911 isin 119864 (5)

The class 119878lowast

119901(120601)was defined and studied by Ali et al [1]They

obtained the Fekete-Szego inequality for functions in the class119878lowast

119901(120601)The class 119878lowast

1(120601) coincideswith the class 119878lowast(120601) discussed

by Ma and Minda [2] Owa [3] introduced a subclass of 119901-valently Bazilevic functions 119867119901(119860 119861 120572 120573) A function 119891 isin

119860119901 is said to be in the class119867119901(119860 119861 120572 120573) if and only if

(1 minus 120573) (119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891 (119911)

119911119901)

120572

≺1 + 119860119911

1 + 119861119911 119911 isin 119864

(6)

where minus1 le 119861 lt 119860 le 1 120572 ge 0 and 0 le 120573 le 1 We now definethe following subclass of analytic functions

Definition 1 Let 120601(119911) be a univalent starlike function withrespect to 1 which maps the unit disk 119864 onto a region in theright half-plane which is symmetric with respect to the real

2 Journal of Applied Mathematics

axis with 120601(0) = 1 and 1206011015840(0) gt 0 A function 119891 isin 119860119901 is in theclass 119881119901119887120572120573(120601) if

1 minus2

119887+2

119887 (1 minus 120573) (

119891 (119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891 (119911)

119911119901)

120572

≺ 120601 (119911)

(7)

where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0

Definition 2 A function 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

≺ 120601 (119911)

(8)

where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0 In other words a func-tion 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if (119891 lowast 119892)(119911) isin

119881119901119887120572120573(120601)We have the following special cases

(i) 119881119901211(120601) coincides with the class 119878lowast119901(120601) introduced

and studied by Ali et al [1](ii) For 119901 = 1 119887 = 2 and 120573 = 1 we have the class 119861120572(120601)

introduced and studied by Ravichandran et al [4](iii) For 119887 = 2 and 120601(119911) = (1 + 119860119911)(1 + 119861119911) the class

1198811199012120572120573(120601) reduces to 119867119901(119860 119861 120572 120573) introduced andstudied by Owa [3]

(iv) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205721(120601)reduces to the class 119861119901(120572 120574) defined as

119861119901 (120572 120574) = 119891 isin 119860119901 Re(1199111198911015840(119911)

119901119891 (119911)(119891 (119911)

119911119901)

120572

) gt 120574

0 le 120574 lt 1 119911 isin 119864

(9)

(v) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205720(120601)is defined as

119891 isin 119860119901 Re(119891 (119911)

119911119901)

120572

gt 120574 0 le 120574 lt 1 119911 isin 119864 (10)

(vi) 1198811201(120601) = 119878lowast(120601) is investigated by Ma and Minda

[2](vii) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 1198811210(120601)

reduces to the class

119861120574 = 119891 isin 1198601 Re119891 (119911)

119911gt 120574 (11)

studied by Chen [5]

We need the following results to obtain our main results

Lemma 3 (see [1]) LetΩ be the class of analytic functions119908normalized by119908(0) = 0 satisfying condition |119908(119911)| lt 1 If119908 isin

Ω and 119908(119911) = 1199081119911 + 11990821199112 + sdot sdot sdot 119911 isin 119864 then

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816le

minus119905 119905 lt minus1

1 minus1 le 119905 le 1

119905 119905 gt 1

(12)

For 119905 le minus1 or 119905 ge 1 the equality holds if and only if 119908(119911) = 119911or one of its rotation For minus1 le 119905 le 1 the equality holds if119908(119911) = 119911

2 or one of its rotation the equality holds for 119905 = minus1if and only if 119908(119911) = 119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or oneof its rotation while for 119905 = 1 the equality holds if and only if119908(119911) = minus119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or one of its rotationThe above upper bound for minus1 lt 119905 lt 1 is sharp and it can beimproved as follows

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816+ (1 + 119905)

1003816100381610038161003816119908110038161003816100381610038162le 1 minus1 lt 119905 le 0

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816+ (1 minus 119905)

1003816100381610038161003816119908110038161003816100381610038162le 1 0 lt 119905 lt 1

(13)

Lemma 4 (see [6 (7) page 10]) If119908 isin Ω and119908(119911) = 1199081119911 +11990821199112+ sdot sdot sdot 119911 isin 119864 then

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816le max 1 |119905| (14)

for any complex number 119905The result is sharp for the functions119908(119911) = 119911

2 or 119908(119911) = 119911

Lemma 5 (see [7]) If119908 isin Ω then for any real number 1199021 and1199022 the following sharp estimate holds

100381610038161003816100381610038161199083 + 119902111990811199082 + 1199022119908

3

1

10038161003816100381610038161003816le 119867 (1199021 1199022) (15)

where

119867(1199021 1199022)

=

1

(1199021 1199022) isin 1198631 cup 1198632100381610038161003816100381611990221003816100381610038161003816

(1199021 1199022) isin7

⋃119896=3

119863119896

2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) (

100381610038161003816100381611990211003816100381610038161003816 + 1

3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)

)

12

(1199021 1199022) isin 1198638 cup 1198639

1

31199022 (

1199022

1minus 4

11990221minus 41199022

)(1199022

1minus 4

3 (1199022 minus 1))

12

(1199021 1199022) isin 11986310 cup 11986311 plusmn2 1

2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1) (

100381610038161003816100381611990211003816100381610038161003816 minus 1

3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)

)

12

(1199021 1199022) isin 11986312

(16)

Journal of Applied Mathematics 3

The extremal function up to the rotations is of the form

119908 (119911) = 1199113 119908 (119911) = 119911

119908 (119911) = 1199080 (119911) =119911 ([(1 minus 120582) 1205762 + 1205821205761] minus 12057611205762119911)

1 minus [(1 minus 120582) 1205761 + 1205821205762] 119911

119908 (119911) = 1199081 (119911) =119911 (1199051 minus 119911)

1 minus 1199051119911

119908 (119911) = 1199082 (119911) =119911 (1199052 + 119911)

1 + 1199052119911

100381610038161003816100381612057611003816100381610038161003816 =

100381610038161003816100381612057621003816100381610038161003816 = 1 1205761 = 1199050 minus 119890

minus11989412057902 (119886 ∓ 119887)

1205762 = minus119890minus11989412057902 (119894119886 plusmn 119887)

119886 = 1199050 cos1205790

2 119887 = radic1 minus 1199052

0sin2

12057902 120582 =

119887 plusmn 119886

2119887

1199050 = (21199022 (119902

2

1+ 2) minus 3119902

2

1

3 (1199022 minus 1) (11990221minus 41199022)

)

12

1199051 = (

100381610038161003816100381611990211003816100381610038161003816 + 1

3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)

)

12

1199052 = (

100381610038161003816100381611990211003816100381610038161003816 minus 1

3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)

)

12

cos1205790

2=1199021

2(1199022 (1199022

1+ 8) minus 2 (119902

2

1+ 2)

21199022 (11990221+ 2) minus 31199022

1

)

(17)

The sets119863119896 119896 = 1 2 12 are defined as follows

1198631 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le

1

2100381610038161003816100381611990221003816100381610038161003816 le 1

1198632 = (1199021 1199022) 1

2le100381610038161003816100381611990211003816100381610038161003816 le 2

4

27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le 1

1198633 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le

1

2100381610038161003816100381611990221003816100381610038161003816 le minus1

1198634 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge

1

2 1199022 le minus

2

3(100381610038161003816100381611990211003816100381610038161003816 + 1)

1198635 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le 2 1199022 ge 1

1198636 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4 1199022 ge

1

12(1199022

1+ 8)

1198637 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4 1199022 ge

2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1)

1198638 = (1199021 1199022) 1

2le100381610038161003816100381611990211003816100381610038161003816 le 2

minus2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le

4

27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1)

1198639 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 2

minus2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

11986310 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le1

12(1199022

1+ 8)

11986311 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)

11990221minus 2

100381610038161003816100381611990211003816100381610038161003816 + 4

11986312 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)

11990221minus 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1)

(18)

2 Main Results

Theorem 6 Let 120601(119911) = 1+1198611119911+11986121199112 +11986131199113 + sdot sdot sdot where 119861119899119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611) minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611) minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φ (119901 120572 120573 120583) =(120572119901 + 2120573) (2120583 + 120572 minus 1)

2(120572119901 + 120573)2

(19)

4 Journal of Applied Mathematics

If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

minus119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 gt 1205902

(20)

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(21)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(22)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(23)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573)119867 (1199021 1199022) (24)

where119867(1199021 1199022) is defined in Lemma 3 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(25)

These results are sharp

Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction

119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)

such that

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 120601 (119908 (119911))

(27)

Now

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2 + (120572 minus 1) 1198862

119901+1 1199112+2 (120572119901 + 3120573)

119887119901

times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+1 1199113+ sdot sdot sdot

(28)

Also we have

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(29)

Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain

119886119901+1 =11988711990111986111199081

2 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(30)

Journal of Applied Mathematics 5

where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that

119886119901+2 minus 1205831198862

119901+1=

1198871199011198611

2 (120572119901 + 2120573)1199082 minus ]119908

2

1 (31)

where

] =1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611 (32)

The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that

119870120601119899 (0) = [119870120601119899]1015840

(0) minus 1 = 0

119865120582 (0) = 1198651015840

120582(0) minus 1 = 0

119866120582 (0) = 1198661015840

120582(0) minus 1 = 0

(33)

with

1 minus2

119887+2

119887(1 minus 120573)(

119870120601119899(119911)

119911119901)

120572

+ 1205731199111198701015840

120601119899(119911)

119901119870120601119899 (119911)(119870120601119899(119911)

119911119901)

120572

≺ 120601 (119911119899minus1)

1 minus2

119887+2

119887(1 minus 120573) (

119865120582(119911)

119911119901)

120572

+ 1205731199111198651015840

120582(119911)

119901119865120582 (119911)(119865120582(119911)

119911119901)

120572

≺ 120601(119911 (119911 + 120582)

1 + 120582119911)

1 minus2

119887+2

119887(1 minus 120573) (

119866120582(119911)

119911119901)

120572

+ 1205731199111198661015840

120582(119911)

119901119866120582 (119911)(119866120582(119911)

119911119901)

120572

≺ 120601(minus119911 (119911 + 120582)

1 + 120582119911)

(34)

It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations

Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]

Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)

Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901(120601)

Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)

21 Application of Theorem 6 tothe Function Defined by Convolutions

Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φlowast(119901 120572 120573 120583) =

(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)

2(120572119901 + 120573)2

(35)

If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2119892119901+2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φlowast(119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

119887119901

2119892119901+2 (120572119901 + 2120573)(1198871199011198612

1

2Φlowast(119901 120572 120573 120583) minus 1198612)

120583 gt 1205902

(36)

6 Journal of Applied Mathematics

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(37)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(38)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φlowast(119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(39)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)

where119867(1199021 1199022) is defined in Lemma 5 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(41)

These results are sharp

Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 120601 (119908 (119911))

(42)

Now

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862

119901+11198922

119901+1 1199112+2

119887119901(120572119901 + 3120573)

times

119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+11198923

119901+1

1199113+ sdot sdot sdot

(43)

Also we obtain

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(44)

Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain

119886119901+1 =11988711990111986111199081

2119892119901+1 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2119892119901+2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(45)

The remaining proof of the theorem is similar to the proof ofTheorem 6

Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)

Journal of Applied Mathematics 7

Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901119892(120601)

Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1

1198922 =Γ (3) Γ (2 minus 120582)

Γ (3 minus 120582)=

2

2 minus 120582

1198923 =Γ (4) Γ (2 minus 120582)

Γ (4 minus 120582)=

6

(2 minus 120582) (3 minus 120582)

1198611 =8

1205872 1198612 =

16

31205872

(46)

the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]

Acknowledgment

The work here is fully supported by LRGSTD2011UKMICT0302

References

[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994

[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000

[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004

[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975

[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981

[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007

[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000

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Stochastic AnalysisInternational Journal of

2 Journal of Applied Mathematics

axis with 120601(0) = 1 and 1206011015840(0) gt 0 A function 119891 isin 119860119901 is in theclass 119881119901119887120572120573(120601) if

1 minus2

119887+2

119887 (1 minus 120573) (

119891 (119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891 (119911)

119911119901)

120572

≺ 120601 (119911)

(7)

where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0

Definition 2 A function 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

≺ 120601 (119911)

(8)

where 0 le 120573 le 1 120572 ge 0 and 119887 gt 0 In other words a func-tion 119891 isin 119860119901 is in the class 119881119901119887120572120573119892(120601) if (119891 lowast 119892)(119911) isin

119881119901119887120572120573(120601)We have the following special cases

(i) 119881119901211(120601) coincides with the class 119878lowast119901(120601) introduced

and studied by Ali et al [1](ii) For 119901 = 1 119887 = 2 and 120573 = 1 we have the class 119861120572(120601)

introduced and studied by Ravichandran et al [4](iii) For 119887 = 2 and 120601(119911) = (1 + 119860119911)(1 + 119861119911) the class

1198811199012120572120573(120601) reduces to 119867119901(119860 119861 120572 120573) introduced andstudied by Owa [3]

(iv) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205721(120601)reduces to the class 119861119901(120572 120574) defined as

119861119901 (120572 120574) = 119891 isin 119860119901 Re(1199111198911015840(119911)

119901119891 (119911)(119891 (119911)

119911119901)

120572

) gt 120574

0 le 120574 lt 1 119911 isin 119864

(9)

(v) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 11988111990121205720(120601)is defined as

119891 isin 119860119901 Re(119891 (119911)

119911119901)

120572

gt 120574 0 le 120574 lt 1 119911 isin 119864 (10)

(vi) 1198811201(120601) = 119878lowast(120601) is investigated by Ma and Minda

[2](vii) For 120601(119911) = (1 + (1 minus 2120574)119911)(1 minus 119911) the class 1198811210(120601)

reduces to the class

119861120574 = 119891 isin 1198601 Re119891 (119911)

119911gt 120574 (11)

studied by Chen [5]

We need the following results to obtain our main results

Lemma 3 (see [1]) LetΩ be the class of analytic functions119908normalized by119908(0) = 0 satisfying condition |119908(119911)| lt 1 If119908 isin

Ω and 119908(119911) = 1199081119911 + 11990821199112 + sdot sdot sdot 119911 isin 119864 then

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816le

minus119905 119905 lt minus1

1 minus1 le 119905 le 1

119905 119905 gt 1

(12)

For 119905 le minus1 or 119905 ge 1 the equality holds if and only if 119908(119911) = 119911or one of its rotation For minus1 le 119905 le 1 the equality holds if119908(119911) = 119911

2 or one of its rotation the equality holds for 119905 = minus1if and only if 119908(119911) = 119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or oneof its rotation while for 119905 = 1 the equality holds if and only if119908(119911) = minus119911((120582 + 119911)(1 + 120582119911)) (0 le 120582 le 1) or one of its rotationThe above upper bound for minus1 lt 119905 lt 1 is sharp and it can beimproved as follows

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816+ (1 + 119905)

1003816100381610038161003816119908110038161003816100381610038162le 1 minus1 lt 119905 le 0

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816+ (1 minus 119905)

1003816100381610038161003816119908110038161003816100381610038162le 1 0 lt 119905 lt 1

(13)

Lemma 4 (see [6 (7) page 10]) If119908 isin Ω and119908(119911) = 1199081119911 +11990821199112+ sdot sdot sdot 119911 isin 119864 then

100381610038161003816100381610038161199082 minus 119905119908

2

1

10038161003816100381610038161003816le max 1 |119905| (14)

for any complex number 119905The result is sharp for the functions119908(119911) = 119911

2 or 119908(119911) = 119911

Lemma 5 (see [7]) If119908 isin Ω then for any real number 1199021 and1199022 the following sharp estimate holds

100381610038161003816100381610038161199083 + 119902111990811199082 + 1199022119908

3

1

10038161003816100381610038161003816le 119867 (1199021 1199022) (15)

where

119867(1199021 1199022)

=

1

(1199021 1199022) isin 1198631 cup 1198632100381610038161003816100381611990221003816100381610038161003816

(1199021 1199022) isin7

⋃119896=3

119863119896

2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) (

100381610038161003816100381611990211003816100381610038161003816 + 1

3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)

)

12

(1199021 1199022) isin 1198638 cup 1198639

1

31199022 (

1199022

1minus 4

11990221minus 41199022

)(1199022

1minus 4

3 (1199022 minus 1))

12

(1199021 1199022) isin 11986310 cup 11986311 plusmn2 1

2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1) (

100381610038161003816100381611990211003816100381610038161003816 minus 1

3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)

)

12

(1199021 1199022) isin 11986312

(16)

Journal of Applied Mathematics 3

The extremal function up to the rotations is of the form

119908 (119911) = 1199113 119908 (119911) = 119911

119908 (119911) = 1199080 (119911) =119911 ([(1 minus 120582) 1205762 + 1205821205761] minus 12057611205762119911)

1 minus [(1 minus 120582) 1205761 + 1205821205762] 119911

119908 (119911) = 1199081 (119911) =119911 (1199051 minus 119911)

1 minus 1199051119911

119908 (119911) = 1199082 (119911) =119911 (1199052 + 119911)

1 + 1199052119911

100381610038161003816100381612057611003816100381610038161003816 =

100381610038161003816100381612057621003816100381610038161003816 = 1 1205761 = 1199050 minus 119890

minus11989412057902 (119886 ∓ 119887)

1205762 = minus119890minus11989412057902 (119894119886 plusmn 119887)

119886 = 1199050 cos1205790

2 119887 = radic1 minus 1199052

0sin2

12057902 120582 =

119887 plusmn 119886

2119887

1199050 = (21199022 (119902

2

1+ 2) minus 3119902

2

1

3 (1199022 minus 1) (11990221minus 41199022)

)

12

1199051 = (

100381610038161003816100381611990211003816100381610038161003816 + 1

3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)

)

12

1199052 = (

100381610038161003816100381611990211003816100381610038161003816 minus 1

3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)

)

12

cos1205790

2=1199021

2(1199022 (1199022

1+ 8) minus 2 (119902

2

1+ 2)

21199022 (11990221+ 2) minus 31199022

1

)

(17)

The sets119863119896 119896 = 1 2 12 are defined as follows

1198631 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le

1

2100381610038161003816100381611990221003816100381610038161003816 le 1

1198632 = (1199021 1199022) 1

2le100381610038161003816100381611990211003816100381610038161003816 le 2

4

27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le 1

1198633 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le

1

2100381610038161003816100381611990221003816100381610038161003816 le minus1

1198634 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge

1

2 1199022 le minus

2

3(100381610038161003816100381611990211003816100381610038161003816 + 1)

1198635 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le 2 1199022 ge 1

1198636 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4 1199022 ge

1

12(1199022

1+ 8)

1198637 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4 1199022 ge

2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1)

1198638 = (1199021 1199022) 1

2le100381610038161003816100381611990211003816100381610038161003816 le 2

minus2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le

4

27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1)

1198639 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 2

minus2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

11986310 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le1

12(1199022

1+ 8)

11986311 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)

11990221minus 2

100381610038161003816100381611990211003816100381610038161003816 + 4

11986312 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)

11990221minus 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1)

(18)

2 Main Results

Theorem 6 Let 120601(119911) = 1+1198611119911+11986121199112 +11986131199113 + sdot sdot sdot where 119861119899119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611) minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611) minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φ (119901 120572 120573 120583) =(120572119901 + 2120573) (2120583 + 120572 minus 1)

2(120572119901 + 120573)2

(19)

4 Journal of Applied Mathematics

If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

minus119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 gt 1205902

(20)

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(21)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(22)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(23)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573)119867 (1199021 1199022) (24)

where119867(1199021 1199022) is defined in Lemma 3 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(25)

These results are sharp

Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction

119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)

such that

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 120601 (119908 (119911))

(27)

Now

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2 + (120572 minus 1) 1198862

119901+1 1199112+2 (120572119901 + 3120573)

119887119901

times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+1 1199113+ sdot sdot sdot

(28)

Also we have

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(29)

Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain

119886119901+1 =11988711990111986111199081

2 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(30)

Journal of Applied Mathematics 5

where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that

119886119901+2 minus 1205831198862

119901+1=

1198871199011198611

2 (120572119901 + 2120573)1199082 minus ]119908

2

1 (31)

where

] =1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611 (32)

The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that

119870120601119899 (0) = [119870120601119899]1015840

(0) minus 1 = 0

119865120582 (0) = 1198651015840

120582(0) minus 1 = 0

119866120582 (0) = 1198661015840

120582(0) minus 1 = 0

(33)

with

1 minus2

119887+2

119887(1 minus 120573)(

119870120601119899(119911)

119911119901)

120572

+ 1205731199111198701015840

120601119899(119911)

119901119870120601119899 (119911)(119870120601119899(119911)

119911119901)

120572

≺ 120601 (119911119899minus1)

1 minus2

119887+2

119887(1 minus 120573) (

119865120582(119911)

119911119901)

120572

+ 1205731199111198651015840

120582(119911)

119901119865120582 (119911)(119865120582(119911)

119911119901)

120572

≺ 120601(119911 (119911 + 120582)

1 + 120582119911)

1 minus2

119887+2

119887(1 minus 120573) (

119866120582(119911)

119911119901)

120572

+ 1205731199111198661015840

120582(119911)

119901119866120582 (119911)(119866120582(119911)

119911119901)

120572

≺ 120601(minus119911 (119911 + 120582)

1 + 120582119911)

(34)

It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations

Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]

Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)

Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901(120601)

Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)

21 Application of Theorem 6 tothe Function Defined by Convolutions

Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φlowast(119901 120572 120573 120583) =

(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)

2(120572119901 + 120573)2

(35)

If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2119892119901+2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φlowast(119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

119887119901

2119892119901+2 (120572119901 + 2120573)(1198871199011198612

1

2Φlowast(119901 120572 120573 120583) minus 1198612)

120583 gt 1205902

(36)

6 Journal of Applied Mathematics

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(37)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(38)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φlowast(119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(39)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)

where119867(1199021 1199022) is defined in Lemma 5 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(41)

These results are sharp

Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 120601 (119908 (119911))

(42)

Now

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862

119901+11198922

119901+1 1199112+2

119887119901(120572119901 + 3120573)

times

119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+11198923

119901+1

1199113+ sdot sdot sdot

(43)

Also we obtain

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(44)

Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain

119886119901+1 =11988711990111986111199081

2119892119901+1 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2119892119901+2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(45)

The remaining proof of the theorem is similar to the proof ofTheorem 6

Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)

Journal of Applied Mathematics 7

Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901119892(120601)

Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1

1198922 =Γ (3) Γ (2 minus 120582)

Γ (3 minus 120582)=

2

2 minus 120582

1198923 =Γ (4) Γ (2 minus 120582)

Γ (4 minus 120582)=

6

(2 minus 120582) (3 minus 120582)

1198611 =8

1205872 1198612 =

16

31205872

(46)

the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]

Acknowledgment

The work here is fully supported by LRGSTD2011UKMICT0302

References

[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994

[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000

[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004

[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975

[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981

[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007

[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000

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Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 3

The extremal function up to the rotations is of the form

119908 (119911) = 1199113 119908 (119911) = 119911

119908 (119911) = 1199080 (119911) =119911 ([(1 minus 120582) 1205762 + 1205821205761] minus 12057611205762119911)

1 minus [(1 minus 120582) 1205761 + 1205821205762] 119911

119908 (119911) = 1199081 (119911) =119911 (1199051 minus 119911)

1 minus 1199051119911

119908 (119911) = 1199082 (119911) =119911 (1199052 + 119911)

1 + 1199052119911

100381610038161003816100381612057611003816100381610038161003816 =

100381610038161003816100381612057621003816100381610038161003816 = 1 1205761 = 1199050 minus 119890

minus11989412057902 (119886 ∓ 119887)

1205762 = minus119890minus11989412057902 (119894119886 plusmn 119887)

119886 = 1199050 cos1205790

2 119887 = radic1 minus 1199052

0sin2

12057902 120582 =

119887 plusmn 119886

2119887

1199050 = (21199022 (119902

2

1+ 2) minus 3119902

2

1

3 (1199022 minus 1) (11990221minus 41199022)

)

12

1199051 = (

100381610038161003816100381611990211003816100381610038161003816 + 1

3 (100381610038161003816100381611990211003816100381610038161003816 + 1 + 1199022)

)

12

1199052 = (

100381610038161003816100381611990211003816100381610038161003816 minus 1

3 (100381610038161003816100381611990211003816100381610038161003816 minus 1 minus 1199022)

)

12

cos1205790

2=1199021

2(1199022 (1199022

1+ 8) minus 2 (119902

2

1+ 2)

21199022 (11990221+ 2) minus 31199022

1

)

(17)

The sets119863119896 119896 = 1 2 12 are defined as follows

1198631 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le

1

2100381610038161003816100381611990221003816100381610038161003816 le 1

1198632 = (1199021 1199022) 1

2le100381610038161003816100381611990211003816100381610038161003816 le 2

4

27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le 1

1198633 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le

1

2100381610038161003816100381611990221003816100381610038161003816 le minus1

1198634 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge

1

2 1199022 le minus

2

3(100381610038161003816100381611990211003816100381610038161003816 + 1)

1198635 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 le 2 1199022 ge 1

1198636 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4 1199022 ge

1

12(1199022

1+ 8)

1198637 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4 1199022 ge

2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1)

1198638 = (1199021 1199022) 1

2le100381610038161003816100381611990211003816100381610038161003816 le 2

minus2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le

4

27(100381610038161003816100381611990211003816100381610038161003816 + 1)3minus (100381610038161003816100381611990211003816100381610038161003816 + 1)

1198639 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 2

minus2

3(100381610038161003816100381611990211003816100381610038161003816 + 1) le 1199022 le

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

11986310 = (1199021 1199022) 2 le100381610038161003816100381611990211003816100381610038161003816 le 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le1

12(1199022

1+ 8)

11986311 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 + 1)

11990221+ 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)

11990221minus 2

100381610038161003816100381611990211003816100381610038161003816 + 4

11986312 = (1199021 1199022) 100381610038161003816100381611990211003816100381610038161003816 ge 4

2100381610038161003816100381611990211003816100381610038161003816 (100381610038161003816100381611990211003816100381610038161003816 minus 1)

11990221minus 2

100381610038161003816100381611990211003816100381610038161003816 + 4

le 1199022 le2

3(100381610038161003816100381611990211003816100381610038161003816 minus 1)

(18)

2 Main Results

Theorem 6 Let 120601(119911) = 1+1198611119911+11986121199112 +11986131199113 + sdot sdot sdot where 119861119899119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611) minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611) minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =(120572119901 + 120573)

2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φ (119901 120572 120573 120583) =(120572119901 + 2120573) (2120583 + 120572 minus 1)

2(120572119901 + 120573)2

(19)

4 Journal of Applied Mathematics

If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

minus119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 gt 1205902

(20)

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(21)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(22)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(23)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573)119867 (1199021 1199022) (24)

where119867(1199021 1199022) is defined in Lemma 3 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(25)

These results are sharp

Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction

119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)

such that

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 120601 (119908 (119911))

(27)

Now

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2 + (120572 minus 1) 1198862

119901+1 1199112+2 (120572119901 + 3120573)

119887119901

times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+1 1199113+ sdot sdot sdot

(28)

Also we have

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(29)

Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain

119886119901+1 =11988711990111986111199081

2 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(30)

Journal of Applied Mathematics 5

where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that

119886119901+2 minus 1205831198862

119901+1=

1198871199011198611

2 (120572119901 + 2120573)1199082 minus ]119908

2

1 (31)

where

] =1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611 (32)

The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that

119870120601119899 (0) = [119870120601119899]1015840

(0) minus 1 = 0

119865120582 (0) = 1198651015840

120582(0) minus 1 = 0

119866120582 (0) = 1198661015840

120582(0) minus 1 = 0

(33)

with

1 minus2

119887+2

119887(1 minus 120573)(

119870120601119899(119911)

119911119901)

120572

+ 1205731199111198701015840

120601119899(119911)

119901119870120601119899 (119911)(119870120601119899(119911)

119911119901)

120572

≺ 120601 (119911119899minus1)

1 minus2

119887+2

119887(1 minus 120573) (

119865120582(119911)

119911119901)

120572

+ 1205731199111198651015840

120582(119911)

119901119865120582 (119911)(119865120582(119911)

119911119901)

120572

≺ 120601(119911 (119911 + 120582)

1 + 120582119911)

1 minus2

119887+2

119887(1 minus 120573) (

119866120582(119911)

119911119901)

120572

+ 1205731199111198661015840

120582(119911)

119901119866120582 (119911)(119866120582(119911)

119911119901)

120572

≺ 120601(minus119911 (119911 + 120582)

1 + 120582119911)

(34)

It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations

Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]

Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)

Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901(120601)

Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)

21 Application of Theorem 6 tothe Function Defined by Convolutions

Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φlowast(119901 120572 120573 120583) =

(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)

2(120572119901 + 120573)2

(35)

If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2119892119901+2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φlowast(119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

119887119901

2119892119901+2 (120572119901 + 2120573)(1198871199011198612

1

2Φlowast(119901 120572 120573 120583) minus 1198612)

120583 gt 1205902

(36)

6 Journal of Applied Mathematics

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(37)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(38)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φlowast(119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(39)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)

where119867(1199021 1199022) is defined in Lemma 5 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(41)

These results are sharp

Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 120601 (119908 (119911))

(42)

Now

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862

119901+11198922

119901+1 1199112+2

119887119901(120572119901 + 3120573)

times

119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+11198923

119901+1

1199113+ sdot sdot sdot

(43)

Also we obtain

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(44)

Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain

119886119901+1 =11988711990111986111199081

2119892119901+1 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2119892119901+2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(45)

The remaining proof of the theorem is similar to the proof ofTheorem 6

Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)

Journal of Applied Mathematics 7

Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901119892(120601)

Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1

1198922 =Γ (3) Γ (2 minus 120582)

Γ (3 minus 120582)=

2

2 minus 120582

1198923 =Γ (4) Γ (2 minus 120582)

Γ (4 minus 120582)=

6

(2 minus 120582) (3 minus 120582)

1198611 =8

1205872 1198612 =

16

31205872

(46)

the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]

Acknowledgment

The work here is fully supported by LRGSTD2011UKMICT0302

References

[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994

[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000

[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004

[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975

[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981

[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007

[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000

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Volume 2014

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Stochastic AnalysisInternational Journal of

4 Journal of Applied Mathematics

If119891(119911) is of the form (1) and belongs to the class119881119901119887120572120573(120601) then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

minus119887119901

2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φ (119901 120572 120573 120583))

120583 gt 1205902

(20)

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(21)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1

1198871199011198611(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583 + 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2 (120572119901 + 2120573)

(22)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(23)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573)119867 (1199021 1199022) (24)

where119867(1199021 1199022) is defined in Lemma 3 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(25)

These results are sharp

Proof Since 119891 isin 119881119901119887120572120573(120601) therefore we have for a Schwarzfunction

119908 (119911) = 1199081119911 + 11990821199112+ sdot sdot sdot 119911 isin 119864 (26)

such that

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 120601 (119908 (119911))

(27)

Now

1 minus2

119887+2

119887(1 minus 120573) (

119891(119911)

119911119901)

120572

+ 1205731199111198911015840(119911)

119901119891 (119911)(119891(119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2 + (120572 minus 1) 1198862

119901+1 1199112+2 (120572119901 + 3120573)

119887119901

times 119886119901+3 + (120572 minus 1) 119886119901+1119886119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+1 1199113+ sdot sdot sdot

(28)

Also we have

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(29)

Comparing the coefficients of 119911 1199112 1199113 and after simple calcu-lations we obtain

119886119901+1 =11988711990111986111199081

2 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(30)

Journal of Applied Mathematics 5

where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that

119886119901+2 minus 1205831198862

119901+1=

1198871199011198611

2 (120572119901 + 2120573)1199082 minus ]119908

2

1 (31)

where

] =1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611 (32)

The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that

119870120601119899 (0) = [119870120601119899]1015840

(0) minus 1 = 0

119865120582 (0) = 1198651015840

120582(0) minus 1 = 0

119866120582 (0) = 1198661015840

120582(0) minus 1 = 0

(33)

with

1 minus2

119887+2

119887(1 minus 120573)(

119870120601119899(119911)

119911119901)

120572

+ 1205731199111198701015840

120601119899(119911)

119901119870120601119899 (119911)(119870120601119899(119911)

119911119901)

120572

≺ 120601 (119911119899minus1)

1 minus2

119887+2

119887(1 minus 120573) (

119865120582(119911)

119911119901)

120572

+ 1205731199111198651015840

120582(119911)

119901119865120582 (119911)(119865120582(119911)

119911119901)

120572

≺ 120601(119911 (119911 + 120582)

1 + 120582119911)

1 minus2

119887+2

119887(1 minus 120573) (

119866120582(119911)

119911119901)

120572

+ 1205731199111198661015840

120582(119911)

119901119866120582 (119911)(119866120582(119911)

119911119901)

120572

≺ 120601(minus119911 (119911 + 120582)

1 + 120582119911)

(34)

It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations

Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]

Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)

Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901(120601)

Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)

21 Application of Theorem 6 tothe Function Defined by Convolutions

Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φlowast(119901 120572 120573 120583) =

(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)

2(120572119901 + 120573)2

(35)

If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2119892119901+2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φlowast(119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

119887119901

2119892119901+2 (120572119901 + 2120573)(1198871199011198612

1

2Φlowast(119901 120572 120573 120583) minus 1198612)

120583 gt 1205902

(36)

6 Journal of Applied Mathematics

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(37)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(38)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φlowast(119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(39)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)

where119867(1199021 1199022) is defined in Lemma 5 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(41)

These results are sharp

Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 120601 (119908 (119911))

(42)

Now

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862

119901+11198922

119901+1 1199112+2

119887119901(120572119901 + 3120573)

times

119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+11198923

119901+1

1199113+ sdot sdot sdot

(43)

Also we obtain

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(44)

Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain

119886119901+1 =11988711990111986111199081

2119892119901+1 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2119892119901+2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(45)

The remaining proof of the theorem is similar to the proof ofTheorem 6

Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)

Journal of Applied Mathematics 7

Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901119892(120601)

Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1

1198922 =Γ (3) Γ (2 minus 120582)

Γ (3 minus 120582)=

2

2 minus 120582

1198923 =Γ (4) Γ (2 minus 120582)

Γ (4 minus 120582)=

6

(2 minus 120582) (3 minus 120582)

1198611 =8

1205872 1198612 =

16

31205872

(46)

the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]

Acknowledgment

The work here is fully supported by LRGSTD2011UKMICT0302

References

[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994

[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000

[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004

[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975

[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981

[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007

[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 5

where 1199021 and 1199022 are defined in (25) It can be easily followedfrom (30) that

119886119901+2 minus 1205831198862

119901+1=

1198871199011198611

2 (120572119901 + 2120573)1199082 minus ]119908

2

1 (31)

where

] =1198871199011198611

2Φ (119901 120572 120573 120583) minus

1198612

1198611 (32)

The results from (20) to (22) are obtained by using Lemma 3(23) by using Lemma 4 and (24) by using Lemma 5 To showthat these results are sharp we define the functions 119870120601119899(119911)119865120582(119911) and 119866120582(119911) such that

119870120601119899 (0) = [119870120601119899]1015840

(0) minus 1 = 0

119865120582 (0) = 1198651015840

120582(0) minus 1 = 0

119866120582 (0) = 1198661015840

120582(0) minus 1 = 0

(33)

with

1 minus2

119887+2

119887(1 minus 120573)(

119870120601119899(119911)

119911119901)

120572

+ 1205731199111198701015840

120601119899(119911)

119901119870120601119899 (119911)(119870120601119899(119911)

119911119901)

120572

≺ 120601 (119911119899minus1)

1 minus2

119887+2

119887(1 minus 120573) (

119865120582(119911)

119911119901)

120572

+ 1205731199111198651015840

120582(119911)

119901119865120582 (119911)(119865120582(119911)

119911119901)

120572

≺ 120601(119911 (119911 + 120582)

1 + 120582119911)

1 minus2

119887+2

119887(1 minus 120573) (

119866120582(119911)

119911119901)

120572

+ 1205731199111198661015840

120582(119911)

119901119866120582 (119911)(119866120582(119911)

119911119901)

120572

≺ 120601(minus119911 (119911 + 120582)

1 + 120582119911)

(34)

It is clear that the functions 119870120601119899 119865120582 119866120582 isin 119881119901119887120572120573(120601) Let119870120601 = 1198701206012 If 120583 lt 1205901 or 120583 gt 1205902 then the equality occursfor the function 119870120601 or one of its rotations For 1205901 lt 120583 lt 1205902the equality is attained if and only if 119891 is 1198701206013 or one ofits rotations When 120583 = 1205901 then the equality holds for thefunction 119865120582 or one of its rotations If 120583 = 1205902 then the equalityis obtained for the function 119866120582 or one of its rotations

Corollary 7 For 119887 = 2 the results from (20) to (24) coincidewith the results proved by Ramachandran et al [8]

Corollary 8 For 119887 = 2 119901 = 1 and 120573 = 1 the results from(20) to (22) coincide with the results obtained by Ravichandranet al [4] for the class 119861120572(120601)

Corollary 9 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(20) to (24) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901(120601)

Corollary 10 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1 the resultsfrom (20) to (22) coincide with the results obtained by Ma andMinda [2] for the class 119878lowast(120601)

21 Application of Theorem 6 tothe Function Defined by Convolutions

Theorem 11 Let 120601(119911) = 1 + 1198611119911 + 11986121199112 + 11986131199113 + sdot sdot sdot where119861119899 119904 are real with 1198611 gt 0 and 1198612 ge 0 Let

1205901 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 minus 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205902 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 2 (1198612 + 1198611)

minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

1205903 =1198922

119901+1

119892119901+2

(120572119901 + 120573)2

11988711990111986121(120572119901 + 2120573)

times 21198612 minus 1198871199011198612

1

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

Φlowast(119901 120572 120573 120583) =

(120572119901 + 2120573) (2120583 (119892119901+2119892119901+1) + 120572 minus 1)

2(120572119901 + 120573)2

(35)

If 119891(119911) is of the form (1) and belongs to the class 119881119901119887120572120573119892(120601)then

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le

119887119901

2119892119901+2 (120572119901 + 2120573)(1198612 minus

1198871199011198612

1

2Φlowast(119901 120572 120573 120583))

120583 lt 1205901

1198871199011198611

2119892119901+2 (120572119901 + 2120573) 1205901 le 120583 le 1205902

119887119901

2119892119901+2 (120572119901 + 2120573)(1198871199011198612

1

2Φlowast(119901 120572 120573 120583) minus 1198612)

120583 gt 1205902

(36)

6 Journal of Applied Mathematics

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(37)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(38)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φlowast(119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(39)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)

where119867(1199021 1199022) is defined in Lemma 5 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(41)

These results are sharp

Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 120601 (119908 (119911))

(42)

Now

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862

119901+11198922

119901+1 1199112+2

119887119901(120572119901 + 3120573)

times

119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+11198923

119901+1

1199113+ sdot sdot sdot

(43)

Also we obtain

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(44)

Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain

119886119901+1 =11988711990111986111199081

2119892119901+1 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2119892119901+2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(45)

The remaining proof of the theorem is similar to the proof ofTheorem 6

Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)

Journal of Applied Mathematics 7

Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901119892(120601)

Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1

1198922 =Γ (3) Γ (2 minus 120582)

Γ (3 minus 120582)=

2

2 minus 120582

1198923 =Γ (4) Γ (2 minus 120582)

Γ (4 minus 120582)=

6

(2 minus 120582) (3 minus 120582)

1198611 =8

1205872 1198612 =

16

31205872

(46)

the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]

Acknowledgment

The work here is fully supported by LRGSTD2011UKMICT0302

References

[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994

[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000

[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004

[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975

[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981

[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007

[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Applied Mathematics

Furthermore for 1205901 le 120583 le 1205903

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 minus

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

+1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(37)

and for 1205903 le 120583 le 1205902

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

+1198922

119901+1

1198871199011198611119892119901+2(2(1 +

1198612

1198611)(120572119901 + 120573)

2

(120572119901 + 2120573)

minus1198871199011198611

2(2120583

119892119901+2

119892119901+1+ 120572 minus 1))

10038161003816100381610038161003816119886119901+1

10038161003816100381610038161003816

2

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)

(38)

For any complex number 120583

10038161003816100381610038161003816119886119901+2 minus 120583119886

2

119901+1

10038161003816100381610038161003816

le1198871199011198611

2119892119901+2 (120572119901 + 2120573)max1

10038161003816100381610038161003816100381610038161003816

1198871199011198611

2Φlowast(119901 120572 120573 120583) minus

1198612

1198611

10038161003816100381610038161003816100381610038161003816

(39)

Also

10038161003816100381610038161003816119886119901+3

10038161003816100381610038161003816le

1198871199011198611

2 (120572119901 + 3120573) 119892119901+3119867(1199021 1199022) (40)

where119867(1199021 1199022) is defined in Lemma 5 and

1199021 =21198612

1198611+1198871199011198611

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

1199022 =1198613

1198611+ (

1198871199011198611

2)

2(120572 minus 1) (2120572 minus 1) (120572119901 + 3120573)

6(120572119901 + 120573)3

+1198871199011198612

2

(1 minus 120572) (120572119901 + 3120573)

(120572119901 + 120573) (120572119901 + 2120573)

(41)

These results are sharp

Proof Since119891 isin 119881119901119887120572120573119892(120601) therefore we have for a Schwarzfunction 119908 such that

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 120601 (119908 (119911))

(42)

Now

1 minus2

119887+2

119887 (1 minus 120573)(

(119891 lowast 119892) (119911)

119911119901)

120572

+ 120573119911(119891 lowast 119892)

1015840(119911)

119901 (119891 lowast 119892) (119911)((119891 lowast 119892) (119911)

119911119901)

120572

= 1 +2

119887119901(120572119901 + 120573) 119886119901+1119892119901+1119911 +

1

119887119901(120572119901 + 2120573)

times 2119886119901+2119892119901+2 + (120572 minus 1) 1198862

119901+11198922

119901+1 1199112+2

119887119901(120572119901 + 3120573)

times

119886119901+3119892119901+3 + (120572 minus 1) 119886119901+1119886119901+2119892119901+1119892119901+2

+(120572 minus 1) (120572 minus 2)

61198863

119901+11198923

119901+1

1199113+ sdot sdot sdot

(43)

Also we obtain

120601 (119908 (119911)) = 1 + 11986111199081119911 + (11986111199082 + 11986111199082

1) 1199112

+ (11986111199083 + 2119861211990811199082 + 11986131199083

1) 1199113+ sdot sdot sdot

(44)

Comparing the coefficients of 119911 1199112 1199113 and after simplecalculations we obtain

119886119901+1 =11988711990111986111199081

2119892119901+1 (120572119901 + 120573)

119886119901+2 =1198871199011198611

2119892119901+2 (120572119901 + 2120573)

times 1199082 minus (1198871199011198611

2

(120572 minus 1) (120572119901 + 2120573)

2(120572119901 + 120573)2

minus1198612

1198611)1199082

1

119886119901+3 =1198871199011198611

2119892119901+3 (120572119901 + 2120573)1199083 + 119902111990811199082 + 1199022119908

3

1

(45)

The remaining proof of the theorem is similar to the proof ofTheorem 6

Corollary 12 For 119887 = 2 the results from (36) to (40) coincidewith the results proved by Ramachandran et al [8] for the class1198771199011120572120573119892(120601)

Journal of Applied Mathematics 7

Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901119892(120601)

Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1

1198922 =Γ (3) Γ (2 minus 120582)

Γ (3 minus 120582)=

2

2 minus 120582

1198923 =Γ (4) Γ (2 minus 120582)

Γ (4 minus 120582)=

6

(2 minus 120582) (3 minus 120582)

1198611 =8

1205872 1198612 =

16

31205872

(46)

the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]

Acknowledgment

The work here is fully supported by LRGSTD2011UKMICT0302

References

[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994

[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000

[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004

[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975

[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981

[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007

[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Applied Mathematics 7

Corollary 13 For 119887 = 2 120572 = 0 and 120573 = 1 the results from(36) to (40) coincide with the results obtained by Ali et al [1]for the class 119878lowast

119901119892(120601)

Corollary 14 For 119887 = 2 119901 = 1 120572 = 0 and 120573 = 1

1198922 =Γ (3) Γ (2 minus 120582)

Γ (3 minus 120582)=

2

2 minus 120582

1198923 =Γ (4) Γ (2 minus 120582)

Γ (4 minus 120582)=

6

(2 minus 120582) (3 minus 120582)

1198611 =8

1205872 1198612 =

16

31205872

(46)

the results from (36) to (38) coincide with the results obtainedby Srivastava and Mishra [9]

Acknowledgment

The work here is fully supported by LRGSTD2011UKMICT0302

References

[1] R M Ali V Ravichandran and N Seenivasagan ldquoCoefficientbounds for 119901-valent functionsrdquo Applied Mathematics and Com-putation vol 187 no 1 pp 35ndash46 2007

[2] W C Ma and D Minda ldquoA unified treatment of some specialclasses of univalent functionsrdquo in Proceedings of the Conferenceon Complex Analysis pp 157ndash169 International Press 1994

[3] S Owa ldquoProperties of certain integral operatorsrdquo SoutheastAsian Bulletin of Mathematics vol 24 no 3 pp 411ndash419 2000

[4] V Ravichandran A Gangadharan and M Darus ldquoFekete-Szego inequality for certain class of Bazilevic functionsrdquo FarEast Journal of Mathematical Sciences vol 15 no 2 pp 171ndash1802004

[5] M P Chen ldquoOn the regular functions satisfying 119877119890119891(119911)119911 gt120588rdquo Bulletin of the Institute of Mathematics Academia Sinica vol3 no 1 pp 65ndash70 1975

[6] F R Keogh andE PMerkes ldquoA coefficient inequality for certainclasses of analytic functionsrdquo Proceedings of the AmericanMathematical Society vol 20 pp 8ndash12 1969

[7] D V Prokhorov and J Szynal ldquoInverse coefficients for(120572 120573)-convex functionsrdquo Annales Universitatis Mariae Curie-Skłodowska A vol 35 pp 125ndash143 1981

[8] C Ramachandran S Sivasubramanian and H SilvermanldquoCertain coefficient bounds for 119901-valent functionsrdquo Interna-tional Journal of Mathematics and Mathematical Sciences vol2007 Article ID 46576 11 pages 2007

[9] H M Srivastava and A K Mishra ldquoApplications of fractionalcalculus to parabolic starlike and uniformly convex functionsrdquoComputers ampMathematics with Applications vol 39 no 3-4 pp57ndash69 2000

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of