Research ArticleSome Integral Inequalities for Harmonically(120572 119904)-Convex Functions

Serap Oumlzcan

Department of Mathematics Faculty of Art and Science Kırklareli University 39100 Kırklareli Turkey

Correspondence should be addressed to Serap Ozcan serapozcannyahoocom

Received 26 February 2019 Accepted 11 July 2019 Published 30 July 2019

Academic Editor Seppo Hassi

Copyright copy 2019 Serap OzcanThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In the paper the author introduces a new class of harmonically convex functions which is called harmonically (120572 119904)-convexfunctions and establishes some new integral inequalities of the Hermite-Hadamard type for harmonically (120572 119904)-convex functionsThe properties of the newly introduced class of harmonically convex functions are also investigated Finally some applications tospecial means are given

1 Introduction

It is well known in the literature that a function 119891 119868 sube R 997888rarrR is said to be convex on interval 119868 if

119891 (119905119909 + (1 minus 119905) 119910) le 119905119891 (x) + (1 minus 119905) 119891 (119910) (1)

for all 119909 119910 isin 119868 and 119905 isin [0 1] If the inequality in (1) holds inthe reverse direction 119891 is said to be concave function

Inequalities play a fundamental part in many branchesof pure and applied sciences A number of studies haveshown that the theory of convex functions has a closelyrelationship with the theory of inequalities One of the mostfamous inequalities for convex functions is named Hermite-Hadamard integral inequality as follows

Suppose 119891 119868 sube R 997888rarr R to be a convex function definedon the interval 119868 of real numbers and 119886 119887 isin 119868 with 119886 lt 119887 thenthe double inequality

119891 (119886 + 1198872 ) le 1

119887 minus 119886 int119887119886

119891 (119909) 119889119909 le 119891 (119886) + 119891 (119887)2 (2)

holds If 119891 is concave function both inequalities in (2) arereversed This inequality provides necessary and sufficientcondition for a function to be convex

In recent years the concept of convexity has beenimproved generalized and extended in many directionssee [1ndash9] Several new classes of convex functions havebeen introduced and new versions of Hermite-Hadamardrsquos

inequality have been obtained In [10] Iscan introducedthe class of harmonically convex functions and investigatedthe Hermite-Hadamard type inequalities for this new classof functions For several recent results generalizationsimprovements and refinements concerning harmonicallyconvex functions see [10ndash16] There have been many studiesdedicated to generalizing the harmonic convex functions andto establishing their Hermite-Hadamard type inequalitiesFor some recent studies onHermite-Hadamard type inequal-ities please refer to the monographs [10ndash12 16ndash24]

The aim of this paper is to introduce the concept ofharmonically (120572 119904)-convex functions and to establish severalnewHermite-Hadamard type inequalities based on these newclass of functions

2 Preliminaries

In this section we recall some basic concepts and results ofharmonically convex functions

Definition 1 ([10]) Let 119868 sub (0 infin) be a real interval A func-tion 119891 119868 997888rarr R is said to be harmonically convex functionif

119891 ( 119909119910119905119909 + (1 minus 119905) 119910) = 119905119891 (119910) + (1 minus 119905) 119891 (119909) (3)

for all 119909 119910 isin 119868 and 119905 isin [0 1]

HindawiJournal of Function SpacesVolume 2019 Article ID 2394021 8 pageshttpsdoiorg10115520192394021

2 Journal of Function Spaces

Definition 2 ([13]) A function 119891 119868 sub (0 infin) 997888rarr R is saidto be harmonically 119904-convex function in second sense where119904 isin (0 1] if

119891 ( 119909119910119905119909 + (1 minus 119905) 119910) = 119905119904119891 (119910) + (1 minus 119905)119904 119891 (119909) (4)

for all 119909 119910 isin 119868 and 119905 isin [0 1]Definition 3 ([15]) A function 119891 119868 sub (0 infin) 997888rarr R is saidto be harmonically 119904-convex function in first sense where 119904 isin[0 1] if

119891 ( 119909119910119905119909 + (1 minus 119905) 119910) = 119905119904119891 (119910) + (1 minus 119905119904) 119891 (119909) (5)

for all 119909 119910 isin 119868 and 119905 isin [0 1]Definition 4 ([12]) The function 119891 (0 119887lowast] 997888rarr R 119887lowast gt 0is said to be harmonically (120572 119898)-convex function where 120572 isin[0 1] and 119898 isin (0 1] if

119891 ( 119898119909119910119898119905119909 + (1 minus 119905) 119910) = 119891 (( 119905

119910 + 1 minus 119905119898119909 )minus1)

le 119905120572119891 (119910) + 119898 (1 minus 119905120572) 119891 (119909)(6)

for all 119909 119910 isin (0 119887lowast] and 119905 isin [0 1]Note that if 119898 = 1 in Definition 4 we have definition of

harmonically 119904-convex function in first sense for 120572 = 119904Definition 5 ([25]) A function 119891 119868 sub (0 infin) 997888rarr R is saidto be harmonically 119875-function if

119891 ( 119909119910119905119909 + (1 minus 119905) 119910) = 119891 (119909) + 119891 (119910) (7)

for all 119909 119910 isin 119868 and 119905 isin [0 1]Theorem 6 ([10]) Let 119868 sub (0 infin) 997888rarr R be harmonicallyconvex and 119886 119887 isin 119868 with 119886 lt 119887 If 119891 isin 119871[119886 119887] then thefollowing inequalities hold

119891 ( 2119886119887119886 + 119887) le 119886119887

119887 minus 119886 int119887119886

119891 (119909)1199092 119889119909 le 119891 (119886) + 119891 (119887)

2 (8)

Theorem 7 ([10]) Let119891 119868 sub (0 infin) 997888rarr R be a differentiablefunction on 119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 isharmonically convex on [119886 119887] for 119902 ge 1 then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 (1205822 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205823 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(9)

where

1205821 = 1119886119887 minus 2

(119887 minus 119886)2 ln((119886 + 119887)24119886119887 ) (10)

1205822 = minus 1119887 (119887 minus 119886) + 3119886 + 119887

(119887 minus 119886)3 ln((119886 + 119887)24119886119887 ) (11)

and

1205823 = 1119886 (119887 minus 119886) + 3119887 + 119886

(119887 minus 119886)3 ln((119886 + 119887)24119886119887 ) (12)

Theorem8 ([10]) Let119891 119868 sub (0 infin) 997888rarr R be a differentiablefunction on 119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902is harmonically convex on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205831 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205832 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(13)

where

1205831 = 1198862minus2119902 + 1198871minus2119902 [(119887 minus 119886) (1 minus 2119902) minus 119886]2 (119887 minus 119886)2 (1 minus 119902) (1 minus 2119902)

1205832 = 1198872minus2119902 minus 1198861minus2119902 [(119887 minus 119886) (1 minus 2119902) + 119887]2 (119887 minus 119886)2 (1 minus 119902) (1 minus 2119902)

(14)

Definition 9 ([9]) Let 119891 and 119892 be two functions If

(119891 (119909) minus 119891 (119910)) (119892 (119909) minus 119892 (119910)) ge 0 (15)

holds for all 119909 119910 isin R then 119891 and 119892 are said to be similarlyordered functions

Definition 10 ([23]) A function 119891 119868 sub R 997888rarr R is said to be(120572 119904)-convex if119891 (119905119909 + (1 minus 119905) 119910) le 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910) (16)

for all 119909 119910 isin 119868 and 119905 isin (0 1) with 119904 isin [minus1 1] and 120572 isin (0 1]In order to prove some of our main results we need the

following lemma

Lemma 11 ([10]) Let 119891 119868 sub (0 infin) 997888rarr R be a differentiablefunction on 119868∘ and 119886 119887 isin 119868 with 119886 lt 119887 If 1198911015840 isin 119871[119886 119887] then

119891 (119886) + 119891 (119887)2 minus 119886119887

119887 minus 119886 int119887119886

119891 (119909)1199092 119889119909 = 119886119887 (119887 minus 119886)

2sdot int10

1 minus 2119905(119905119887 + (1 minus 119905) 119886)2119891

1015840 ( 119886119887119905119887 + (1 minus 119905) 119886) 119889119905

(17)

3 Main Results

In this section we define the concept of harmonically (120572 119904)-convex functions and derive our main results Throughoutthis section 119868 sub (0 infin) is the interval and 119868∘ is the interiorof 119868

Journal of Function Spaces 3

Definition 12 A function 119891 119868 997888rarr R is said to be harmon-ically (120572 119904)-convex where (120572 119904) isin (0 1]2 if

119891 ( 119909119910119905119910 + (1 minus 119905) 119909) = 119891 (( 119905

119909 + 1 minus 119905119910 )minus1)

le 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910)(18)

for all 119909 119910 isin 119868 and 119905 isin (0 1) If the inequality in (18) isreversed then 119891 is said to be harmonically (120572 119904)-concave

Now we give some special cases of our proposed defini-tion of harmonically (120572 119904)-convex functions

(I) If 120572 = 1 then we have the definition of harmonically119904-convex functions in second sense(II) If 119904 = 1 then we have the definition of harmonically(120572 1)-convex functions or in other words harmonically 119904-

convex functions in first sense by writing 119904 instead of 120572(III) If 120572 = 119904 = 1 then we have the definition of

harmonically convex functions Thus every harmonicallyconvex function is also harmonically (1 1)-convex function

The following proposition is obvious

Proposition 13 Let 119891 119868 sub (0 infin) 997888rarr R be a function

(1) If119891 is (120572 119904)-convex and nondecreasing function then119891 is harmonically (120572 119904)-convex(2) If 119891 is harmonically (120572 119904)-convex and nonincreasing

function then 119891 is (120572 119904)-convexProof For all 119905 isin [0 1] and 119909 119910 isin 119868 we have

119905 (1 minus 119905) (119909 minus 119910)2 ge 0 (19)

So the following inequality holds119909119910

119905119910 + (1 minus 119905) 119909 le 119905119909 + (1 minus 119905) 119910 (20)

By inequality (20) the proof is completed

Remark 14 According to Proposition 13 every nondecreas-ing convex function (or (1 1)-convex function) is also har-monically convex function (or harmonically (1 1)-convexfunction)

Example 15 The function 119891 (0 infin) 997888rarr R 119891(119909) =119909 is a nondecreasing (1 1)-convex function According toRemark 14 119891 is harmonically (1 1)-convex function (orharmonically (120572 119904)-convex function for 120572 = 119904 = 1)Proposition 16 Let 119891 and 119892 be two harmonically (120572 119904)-convex functions If 119891 and 119892 are similarly ordered functionsand 119905120572119904+(1minus119905120572)119904 le 1 then the product119891119892 is also harmonically(120572 119904)-convex functionProof Let 119891 and 119892 be harmonically (120572 119904)-convex functionsThen

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) 119892 ( 119886119887

119905119887 + (1 minus 119905) 119886)le [119905120572119904119891 (119886) + (1 minus 119905120572)119904 119891 (119887)]

sdot [119905120572119904119892 (119886) + (1 minus 119905120572)119904 119892 (119887)] = 1199052120572119904119891 (119886) 119892 (119886)+ 119905120572119904 (1 minus 119905120572)119904 [119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886)]+ (1 minus 119905120572)2119904 119891 (119887) 119892 (119887) le 1199052120572119904119891 (119886) 119892 (119886)+ 119905120572119904 (1 minus 119905120572)119904 [119891 (119886) 119892 (119886) + 119891 (119887) 119892 (119887)]+ (1 minus 119905120572)2119904 119891 (119887) 119892 (119887)= [119905120572119904119891 (119886) 119892 (119886) + (1 minus 119905120572)119904 119891 (119887) 119892 (119887)]sdot [119905120572119904 + (1 minus 119905120572)119904] le 119905120572119904119891 (119886) 119892 (119886) + (1 minus 119905120572)119904 119891 (119887)sdot 119892 (119887)

(21)

Proposition 17 Let 119891 [119886 119887] sub (0 infin) 997888rarr R and119892 [1119887 1119886] 997888rarr R defined by 119892(119909) = 1119909 then 119891 isharmonically (120572 119904)-convex on [119886 119887] where 120572 isin (0 1] 119904 isin[0 1] if and only if 119892 is (120572 119904)-convex on [1119887 1119886]Proof Since

(119891 ∘ 119892) (119905119909 + (1 minus 119905) 119910) = 119891 ( 1119905119909 + (1 minus 119905) 119910)

= 119891 ( 1119905119903 + (1 minus 119905) 119896) = 119891 (( 119905

119903 + 1 minus 119905119896 )minus1)

le 119905120572119904119891 (119903) + (1 minus 119905120572)119904 119891 (119896)= 119905120572119904 (119891 ∘ 119892) (119909) + (1 minus 119905120572)119904 (119891 ∘ 119892) (119910)

(22)

for all 119909 119910 isin [1119887 1119886] 119905 isin [0 1] where 119903 119896 isin [119886 119887] 119909 = 1119903119910 = 1119896This demonstrates necessary condition is provided Now

let us show the sufficient condition is also providedFor all 119909 119910 isin [119886 119887] and 119905 isin [0 1] we have119891 (( 119905

119909 + 1 minus 119905119910 )minus1) = (119891 ∘ 119892) (119905 1119909 + (1 minus 119905) 1

119910)

le 119905120572119904 (119891 ∘ 119892) ( 1119909)

+ (1 minus 119905120572)119904 (119891 ∘ 119892) ( 1119910)

= 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910)

(23)

This completes the proof

Theorem 18 Let 119891 119868 997888rarr R be harmonically (120572 119904)-convexfunction with 120572 isin (0 1] 119904 isin [0 1] If 0 lt 119886 lt 119887 lt infin and119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909

le 119891 (119886) + 119891 (119887)2 int1

0[119905120572119904 + (1 minus 119905120572)119904] 119889119905

(24)

4 Journal of Function Spaces

Proof Since 119891 is a harmonically (120572 119904)-convex function wehave for all 119909 119910 isin 119868

119891 ( 119909119910119905119910 + (1 minus 119905) 119909) le 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910) (25)

which gives

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) le 119905120572119904119891 (119886) + (1 minus 119905120572)119904 119891 (119887)

119891 ( 119886119887119905119886 + (1 minus 119905) 119887) le 119905120572119904119891 (119887) + (1 minus 119905120572)119904 119891 (119886)

(26)

for all 119905 isin [0 1]Adding the above inequalities we have

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) + 119891 ( 119886119887

119905119886 + (1 minus 119905) 119887)le [119891 (119886) + 119891 (119887)] [119905120572119904 + (1 minus 119905120572)119904]

(27)

Integrating the above inequality on [0 1] we obtainint10

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) 119889119905 + int1

0119891 ( 119886119887

119905119886 + (1 minus 119905) 119887) 119889119905

le [119891 (119886) + 119891 (119887)] int10

[119905120572119904 + (1 minus 119905120572)119904] 119889119905(28)

which implies

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909

le 119891 (119886) + 119891 (119887)2 int1

0[119905120572119904 + (1 minus 119905120572)119904] 119889119905

(29)

which is the required result

Remark 19 If we take 120572 = 119904 = 1 in Theorem 18 theninequality (24) becomes the right-hand side of inequality (2)

If we take 120572 = 1 in Theorem 18 we have the followingresult for harmonically 119904-convex functions in second sense

Corollary 20 Let 119891 119868 997888rarr R be harmonically 119904-convexfunction in second sense where 119886 119887 isin 119868 with 119886 lt 119887 and119904 isin [0 1] If 119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909 le 119891 (119886) + 119891 (119887)

119904 + 1 (30)

If we take 119904 = 0 then Theorem 18 collapses to thefollowing result

Corollary 21 Let 119891 119868 997888rarr R be harmonically 119875-functionwhere 119886 119887 isin 119868 with 119886 lt 119887 If 119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909 le 119891 (119886) + 119891 (119887) (31)

Our coming result is the Hermite-Hadamard inequalityfor product of two harmonically (120572 119904)-convex functions

Theorem 22 Let 119891 119892 119868 997888rarr R be two harmonically (120572 119904)-convex functions where 119886 119887 isin 119868 with 119886 lt 119887 120572 isin (0 1] 119904 isin[0 1] If 119891119892 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 le 12057811198721 + 12057821198722 + 12057831198723 (32)

where

1205781 = 119891 (119886) 119892 (119886) (33)

1205782 = 119891 (119887) 119892 (119887) (34)

1205783 = 119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886) (35)

and

1198721 = int10

1199052120572119904119889119905 = 12120572119904 + 1

1198722 = int10

(1 minus 119905120572)2119904 119889119905 = 120573 (2119904 + 1 1120572 + 1)

1198723 = int10

119905120572119904 (1 minus 119905120572)119904 119889119905

(36)

120573 is Euler Beta function defined by

120573 (119909 119910) = int10

119905119909minus1 (1 minus 119905)119910minus1 119889119905 119909 119910 gt 0 (37)

and 21198651 is hypergeometric function defined by

21198651 (119886 119887 119888 119911)= 1

120573 (119887 119888 minus 119887) int10

119905119887minus1 (1 minus 119905)119888minus119887minus1 (1 minus 119911119905)minus119886 119889119905119888 gt 119887 gt 0 |119911| lt 1

(38)

Proof Let 119891 119892 119868 997888rarr R be harmonically (120572 119904)-convexfunctions then we have

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 = int10

119891 ( 119909119910119905119910 + (1 minus 119905) 119909)

sdot 119892 ( 119909119910119905119910 + (1 minus 119905) 119909) 119889119905 le int1

0[119905120572119904119891 (119886)

+ (1 minus 119905120572)119904 119891 (119887)] [119905120572119904119892 (119886) + (1 minus 119905120572)119904 119892 (119887)] 119889119905= int10

(1199052120572119904119891 (119886) 119892 (119886) + (1 minus 119905120572)2119904 119891 (119887) 119892 (119887)+ 119905120572119904 (1 minus 119905120572)119904 [119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886)]) 119889119905= 1205781 int

1

01199052120572119904119889119905 + 1205782 int

1

0(1 minus 119905120572)2119904 119889119905 + 1205783 int

1

0119905120572119904 (1

minus 119905120572)119904 119889119905 = 12057811198721 + 12057821198722 + 12057831198723

(39)

This completes the proof

Journal of Function Spaces 5

Theorem 23 Under the conditions of Theorem 22 if 119891 and 119892are similarly ordered functions then we have

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 le 12057811198731 + 12057821198732 (40)

where 1205781 and 1205782 are given by (33) and (34)

1198731 = int10

119905120572119904119889119905 = 1120572119904 + 1

1198732 = int10

(1 minus 119905120572)119904 119889119905 = 120573 (119904 + 1 1120572 + 1)

(41)

Proof Integrating inequality (21) completes the proof

Theorem 24 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119902 ge 1 with 120572 isin (0 1] and 119904 isin [0 1]then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(42)

where 1205821 is given by (10)

1205811 = int10

|1 minus 2119905| 119905120572119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (43)

and

1205812 = int10

|1 minus 2119905| (1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (44)

Proof Using Lemma 11 and power mean inequality we have

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816 le 119886119887 (119887 minus 119886)

2sdot int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times (int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816

sdot 100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times ( |1 minus 2119905|(119905119887 + (1 minus 119905) 119886)2 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1 minus 119905120572)119904

sdot 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2

sdot 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902 (45)

This completes the proof

Corollary 25 Under the conditions of Theorem 24 if 119902 = 1then we have100381610038161003816100381610038161003816100381610038161003816

119891 (119886) + 119891 (119887)2 minus 119886119887

119887 minus 119886 int119887119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 (1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816)

(46)

where 1205811 and 1205812 are given by (43) and (44) respectively

Remark 26 If we take 120572 = 119904 = 1 in Theorem 24 theninequality (42) becomes inequality (9) of Theorem 7

If we take 120572 = 1 in Theorem 24 then we have result forharmonically 119904-convex functions in second sense

Corollary 27 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in second sense on [119886 119887] for 119902 gt 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205882 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(47)

where 1205821 is given by (10)

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (48)

and

1205882 = int10

|1 minus 2119905| (1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (49)

respectively

If we take 119904 = 1 in Theorem 24 then we have followingresult for harmonically 119904-convex functions in first sense for120572 = 119904Corollary 28 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in first sense then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 120588lowast 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(50)

6 Journal of Function Spaces

where 1205821 is given by (10) and

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (51)

and

120588lowast = 1205821 minus 1205881 (52)

If we take 119904 = 0 in Theorem 24 we have the followingresult for harmonically 119875 -functions

Corollary 29 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function for 119902 ge 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821 [100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(53)

where 1205821 is given by (10)

Theorem 30 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 with 120572 isin(0 1] 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(54)

where

1205951 = int10

119905120572119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (55)

and

1205952 = int10

(1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (56)

respectively

Proof From Lemma 11 Holderrsquos inequality and the fact that|1198911015840|119902 is harmonically (120572 119904) -convex function on [119886 119887] we have100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0|1

minus 2119905|119901 119889119905)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902

100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1

minus 119905120572)119904 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(57)

where

int10

|1 minus 2119905|119901 119889119905 = 1119901 + 1 (58)

This completes the proof

Remark 31 If we take 120572 = 119904 = 1 in Theorem 30 theninequality (54) becomes inequality (13) of Theorem 8

If we take 120572 = 1 in Theorem 30 then we have result forharmonically 119904-convex functions in second sense

Corollary 32 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in second sense on [119886 119887] for 119901 119902 gt 1 1119901 +1119902 = 1 with 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205931 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205932 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(59)

where

1205931 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (60)

and

1205932 = int10

(1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (119904 + 1 1)1198872119902 21198651 (2119902 119904 + 1 119904 + 2 1 minus 119886

119887 ) (61)

respectively

If we take 119904 = 1 inTheorem 30 thenwe have the followingresult for harmonically 119904-convex functions in first sense for120572 = 119904

Journal of Function Spaces 7

Corollary 33 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in first sense on [119886 119887] for 119901 119902 gt 1 1119901+1119902 = 1 with119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205991 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205992 100381610038161003816100381610038161198911015840 (b)10038161003816100381610038161003816119902)1119902

(62)

where

1205991 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (63)

and

1205992 = int10

1 minus 119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) minus 1205991 (64)

respectively

If 119904 = 0 in Theorem 30 then we have result forharmonically 119875-functionsCorollary 34 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot 1205921119902 (100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(65)

where

120592 = int10

1(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) (66)

4 Some Applications to Special Means

Let us recall the following special means for arbitrary realnumbers 119886 and 119887 (119886 = 119887)(1) The arithmetic mean

119860 fl 119860 (119886 119887) = 119886 + 1198872 (67)

(2) The geometric mean

119866 fl 119866 (119886 119887) = radic119886119887 (68)

(3) The 119901-logarithmic mean

119871119901 fl 119871119901 (119886 119887) = [ 119887119901+1 minus 119886119901+1(119901 + 1) (119887 minus 119886)]

1119901

119901 isin R minus1 0

(69)

Proposition 35 Let 0 lt 119886 lt 119887 119902 gt 1 and 119904 isin (0 1) Then wehave

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 1205821minus11199021 (1205881119886119904 + 1205882119887119904)1119902(70)

where 1205821 1205881 and 1205882 are given by (10) (48) and (49)respectively

Proof The assertion follows from inequality (47) in Corol-lary 27 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)Proposition 36 Let 0 lt 119886 lt 119887 119901 gt 1 119902 = 119901(119901 minus 1) and119904 isin (0 1) Then we have

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 ( 1119901 + 1)1119901 (1205991119886119904 + 1205992119887119904)1119902

(71)

where 1205991 and 1205992 are given by (63) and (64)

Proof The assertion follows from inequality (62) in Corol-lary 33 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)5 Conclusion

In this paper we introduce and investigate a new class ofharmonically convex functions which is called harmoni-cally (120572 119904)-convex functions Some new results of Hermite-Hadamard type for the newly introduced class of harmoni-cally convex functions are established Some applications ofthese results to special means have also been presented Theresults of this paper may stimulate further research for theresearchers working in this field

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

References

[1] GDAndersonMKVamanamurthy andMVuorinen ldquoGen-eralized convexity and inequalitiesrdquo Journal of MathematicalAnalysis and Applications vol 335 no 2 pp 1294ndash1308 2007

8 Journal of Function Spaces

[2] W W Breckner ldquoStetigkeitsaussagen Fur Eine Klasse Verall-gemeinerter Konvexer Funktionen In Topologischen LinearenRaumenrdquo Publications de lrsquoInstitutMathematique vol 23 no 37pp 13ndash20 1978

[3] G Critescu and L Lupsa Non-Connected Convexities andApplications KluwerAcademic Publishers DordrechtHolland2002

[4] V G Mihesan ldquoA generalization of the convexityrdquo in Proceed-ings of the Seminer on Functional Equations Approximation andConvexity Cluj-Napoca Romania 1993

[5] C P Niculescu and L E Persson Convex Functions And TheirApplications Springer New York NY USA 2006

[6] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000

[7] C P Niculescu ldquoConvexity according to meansrdquoMathematicalInequalities amp Applications vol 6 no 4 pp 571ndash579 2003

[8] J Pecaricrsquo F Proschan and Y L TongConvex Functions PartialOrderings and Statistical Applications Academic Press NewYork NY USA 1992

[9] S Varosanec ldquoOn h-convexityrdquo Journal of Mathematical Analy-sis and Applications vol 326 no 1 pp 303ndash311 2007

[10] I Iscan ldquoHermite-Hadamard type inequalities for harmonicallyconvex functionsrdquoHacettepe Journal of Mathematics and Statis-tics vol 43 no 6 pp 935ndash942 2014

[11] I A Baloch and I Iscan ldquoSome Hermite-Hadamard typeinequalities for harmonically (sm)-convex functions in secondsenserdquo 2016 httpsarxivorgabs160408445

[12] I Iscan ldquoHermite-Hadamard type inequalities for harmonically(120572m)-convex functionsrdquoHacettepe Journal of Mathematics andStatistics vol 45 no 2 pp 381ndash390 2016

[13] I Iscan ldquoOstrowski type inequalities for harmonically s-convexfunctionsrdquo Konuralp Journal of Mathematics vol 3 no 1 pp63ndash74 2015

[14] M A Noor K I Noor and M U Awan ldquoSome characteri-zations of harmonically log-convex functionsrdquo Proceedings ofthe Jangjeon Mathematical Society Memoirs of the JangjeonMathematical Society vol 17 no 1 pp 51ndash61 2014

[15] M A Noor K I Noor M U Awan and S Costache ldquoSomeintegral inequalities for harmonically h-convex functionsrdquoldquoPolitehnicardquo University of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 77 no 1 pp 5ndash16 2015

[16] T-Y Zhang A-P Ji and FQi ldquoIntegral inequalities ofHermite-HADamard type for harmonically quasi-convex functionsrdquoProceedings of the JangjeonMathematical SocietyMemoirs of theJangjeon Mathematical Society vol 16 no 3 pp 399ndash407 2013

[17] M W Alomari M Darus and U S Kirmaci ldquoSome inequal-ities of Hermite-Hadamard type for s-convex functionsrdquo ActaMathematica Scientia vol 31 no 4 pp 1643ndash1652 2011

[18] S S Dragomir ldquoHermite-Hadamardrsquos type inequalities forconvex functions of selfadjoint operators in HILbert spacesrdquoLinear Algebra and its Applications vol 436 no 5 pp 1503ndash15152012

[19] S S Dragomir and C E M Pearce Selected Topics on Hermite-Hadamard Inequalities and Applications Victoria UniversityFootscray Australia 2000

[20] S S Dragomir J Pecaric and L E Persson ldquoSome inequalitiesof Hadamard typerdquo Soochow Journal of Mathematics vol 21 no3 pp 335ndash341 1995

[21] H Hudzik and L Maligranda ldquoSome remarks on s-convexfunctionsrdquo Aequationes Mathematicae vol 48 no 1 pp 100ndash111 1994

[22] I Iscan ldquoHermite-Hadamard type inequalities for GA-s-convexfunctionsrdquo Le Matematiche vol 69 no 2 pp 129ndash146 2014

[23] B Y Xi D D Gao and F Qi ldquoIntegral Inequalities ofHermitendashHadamard Type for (120572 s)-Convex And (120572 s m)-Convex Functionsrdquo in HAL Archives p 9 2018 httpshalarchives-ouvertesfrhal-01761678

[24] T-Y Zhang A-P Ji and F Qi ldquoSome inequalities of Hermite-HADamard type for GA-convex functions with applications tomeansrdquo Le Matematiche vol 68 no 1 pp 229ndash239 2013

[25] I Iscan S Numan and K Bekar ldquoHermite-Hadamard andSimpson type inequalities for differentiable harmonically P-convex functionsrdquo British Journal of Mathematics amp ComputerScience vol 4 no 14 pp 1908ndash1920 2014

Journal of Function Spaces 3

Definition 12 A function 119891 119868 997888rarr R is said to be harmon-ically (120572 119904)-convex where (120572 119904) isin (0 1]2 if

119891 ( 119909119910119905119910 + (1 minus 119905) 119909) = 119891 (( 119905

119909 + 1 minus 119905119910 )minus1)

le 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910)(18)

for all 119909 119910 isin 119868 and 119905 isin (0 1) If the inequality in (18) isreversed then 119891 is said to be harmonically (120572 119904)-concave

Now we give some special cases of our proposed defini-tion of harmonically (120572 119904)-convex functions

(I) If 120572 = 1 then we have the definition of harmonically119904-convex functions in second sense(II) If 119904 = 1 then we have the definition of harmonically(120572 1)-convex functions or in other words harmonically 119904-

convex functions in first sense by writing 119904 instead of 120572(III) If 120572 = 119904 = 1 then we have the definition of

harmonically convex functions Thus every harmonicallyconvex function is also harmonically (1 1)-convex function

The following proposition is obvious

Proposition 13 Let 119891 119868 sub (0 infin) 997888rarr R be a function

(1) If119891 is (120572 119904)-convex and nondecreasing function then119891 is harmonically (120572 119904)-convex(2) If 119891 is harmonically (120572 119904)-convex and nonincreasing

function then 119891 is (120572 119904)-convexProof For all 119905 isin [0 1] and 119909 119910 isin 119868 we have

119905 (1 minus 119905) (119909 minus 119910)2 ge 0 (19)

So the following inequality holds119909119910

119905119910 + (1 minus 119905) 119909 le 119905119909 + (1 minus 119905) 119910 (20)

By inequality (20) the proof is completed

Remark 14 According to Proposition 13 every nondecreas-ing convex function (or (1 1)-convex function) is also har-monically convex function (or harmonically (1 1)-convexfunction)

Example 15 The function 119891 (0 infin) 997888rarr R 119891(119909) =119909 is a nondecreasing (1 1)-convex function According toRemark 14 119891 is harmonically (1 1)-convex function (orharmonically (120572 119904)-convex function for 120572 = 119904 = 1)Proposition 16 Let 119891 and 119892 be two harmonically (120572 119904)-convex functions If 119891 and 119892 are similarly ordered functionsand 119905120572119904+(1minus119905120572)119904 le 1 then the product119891119892 is also harmonically(120572 119904)-convex functionProof Let 119891 and 119892 be harmonically (120572 119904)-convex functionsThen

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) 119892 ( 119886119887

119905119887 + (1 minus 119905) 119886)le [119905120572119904119891 (119886) + (1 minus 119905120572)119904 119891 (119887)]

sdot [119905120572119904119892 (119886) + (1 minus 119905120572)119904 119892 (119887)] = 1199052120572119904119891 (119886) 119892 (119886)+ 119905120572119904 (1 minus 119905120572)119904 [119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886)]+ (1 minus 119905120572)2119904 119891 (119887) 119892 (119887) le 1199052120572119904119891 (119886) 119892 (119886)+ 119905120572119904 (1 minus 119905120572)119904 [119891 (119886) 119892 (119886) + 119891 (119887) 119892 (119887)]+ (1 minus 119905120572)2119904 119891 (119887) 119892 (119887)= [119905120572119904119891 (119886) 119892 (119886) + (1 minus 119905120572)119904 119891 (119887) 119892 (119887)]sdot [119905120572119904 + (1 minus 119905120572)119904] le 119905120572119904119891 (119886) 119892 (119886) + (1 minus 119905120572)119904 119891 (119887)sdot 119892 (119887)

(21)

Proposition 17 Let 119891 [119886 119887] sub (0 infin) 997888rarr R and119892 [1119887 1119886] 997888rarr R defined by 119892(119909) = 1119909 then 119891 isharmonically (120572 119904)-convex on [119886 119887] where 120572 isin (0 1] 119904 isin[0 1] if and only if 119892 is (120572 119904)-convex on [1119887 1119886]Proof Since

(119891 ∘ 119892) (119905119909 + (1 minus 119905) 119910) = 119891 ( 1119905119909 + (1 minus 119905) 119910)

= 119891 ( 1119905119903 + (1 minus 119905) 119896) = 119891 (( 119905

119903 + 1 minus 119905119896 )minus1)

le 119905120572119904119891 (119903) + (1 minus 119905120572)119904 119891 (119896)= 119905120572119904 (119891 ∘ 119892) (119909) + (1 minus 119905120572)119904 (119891 ∘ 119892) (119910)

(22)

for all 119909 119910 isin [1119887 1119886] 119905 isin [0 1] where 119903 119896 isin [119886 119887] 119909 = 1119903119910 = 1119896This demonstrates necessary condition is provided Now

let us show the sufficient condition is also providedFor all 119909 119910 isin [119886 119887] and 119905 isin [0 1] we have119891 (( 119905

119909 + 1 minus 119905119910 )minus1) = (119891 ∘ 119892) (119905 1119909 + (1 minus 119905) 1

119910)

le 119905120572119904 (119891 ∘ 119892) ( 1119909)

+ (1 minus 119905120572)119904 (119891 ∘ 119892) ( 1119910)

= 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910)

(23)

This completes the proof

Theorem 18 Let 119891 119868 997888rarr R be harmonically (120572 119904)-convexfunction with 120572 isin (0 1] 119904 isin [0 1] If 0 lt 119886 lt 119887 lt infin and119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909

le 119891 (119886) + 119891 (119887)2 int1

0[119905120572119904 + (1 minus 119905120572)119904] 119889119905

(24)

4 Journal of Function Spaces

Proof Since 119891 is a harmonically (120572 119904)-convex function wehave for all 119909 119910 isin 119868

119891 ( 119909119910119905119910 + (1 minus 119905) 119909) le 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910) (25)

which gives

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) le 119905120572119904119891 (119886) + (1 minus 119905120572)119904 119891 (119887)

119891 ( 119886119887119905119886 + (1 minus 119905) 119887) le 119905120572119904119891 (119887) + (1 minus 119905120572)119904 119891 (119886)

(26)

for all 119905 isin [0 1]Adding the above inequalities we have

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) + 119891 ( 119886119887

119905119886 + (1 minus 119905) 119887)le [119891 (119886) + 119891 (119887)] [119905120572119904 + (1 minus 119905120572)119904]

(27)

Integrating the above inequality on [0 1] we obtainint10

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) 119889119905 + int1

0119891 ( 119886119887

119905119886 + (1 minus 119905) 119887) 119889119905

le [119891 (119886) + 119891 (119887)] int10

[119905120572119904 + (1 minus 119905120572)119904] 119889119905(28)

which implies

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909

le 119891 (119886) + 119891 (119887)2 int1

0[119905120572119904 + (1 minus 119905120572)119904] 119889119905

(29)

which is the required result

Remark 19 If we take 120572 = 119904 = 1 in Theorem 18 theninequality (24) becomes the right-hand side of inequality (2)

If we take 120572 = 1 in Theorem 18 we have the followingresult for harmonically 119904-convex functions in second sense

Corollary 20 Let 119891 119868 997888rarr R be harmonically 119904-convexfunction in second sense where 119886 119887 isin 119868 with 119886 lt 119887 and119904 isin [0 1] If 119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909 le 119891 (119886) + 119891 (119887)

119904 + 1 (30)

If we take 119904 = 0 then Theorem 18 collapses to thefollowing result

Corollary 21 Let 119891 119868 997888rarr R be harmonically 119875-functionwhere 119886 119887 isin 119868 with 119886 lt 119887 If 119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909 le 119891 (119886) + 119891 (119887) (31)

Our coming result is the Hermite-Hadamard inequalityfor product of two harmonically (120572 119904)-convex functions

Theorem 22 Let 119891 119892 119868 997888rarr R be two harmonically (120572 119904)-convex functions where 119886 119887 isin 119868 with 119886 lt 119887 120572 isin (0 1] 119904 isin[0 1] If 119891119892 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 le 12057811198721 + 12057821198722 + 12057831198723 (32)

where

1205781 = 119891 (119886) 119892 (119886) (33)

1205782 = 119891 (119887) 119892 (119887) (34)

1205783 = 119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886) (35)

and

1198721 = int10

1199052120572119904119889119905 = 12120572119904 + 1

1198722 = int10

(1 minus 119905120572)2119904 119889119905 = 120573 (2119904 + 1 1120572 + 1)

1198723 = int10

119905120572119904 (1 minus 119905120572)119904 119889119905

(36)

120573 is Euler Beta function defined by

120573 (119909 119910) = int10

119905119909minus1 (1 minus 119905)119910minus1 119889119905 119909 119910 gt 0 (37)

and 21198651 is hypergeometric function defined by

21198651 (119886 119887 119888 119911)= 1

120573 (119887 119888 minus 119887) int10

119905119887minus1 (1 minus 119905)119888minus119887minus1 (1 minus 119911119905)minus119886 119889119905119888 gt 119887 gt 0 |119911| lt 1

(38)

Proof Let 119891 119892 119868 997888rarr R be harmonically (120572 119904)-convexfunctions then we have

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 = int10

119891 ( 119909119910119905119910 + (1 minus 119905) 119909)

sdot 119892 ( 119909119910119905119910 + (1 minus 119905) 119909) 119889119905 le int1

0[119905120572119904119891 (119886)

+ (1 minus 119905120572)119904 119891 (119887)] [119905120572119904119892 (119886) + (1 minus 119905120572)119904 119892 (119887)] 119889119905= int10

(1199052120572119904119891 (119886) 119892 (119886) + (1 minus 119905120572)2119904 119891 (119887) 119892 (119887)+ 119905120572119904 (1 minus 119905120572)119904 [119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886)]) 119889119905= 1205781 int

1

01199052120572119904119889119905 + 1205782 int

1

0(1 minus 119905120572)2119904 119889119905 + 1205783 int

1

0119905120572119904 (1

minus 119905120572)119904 119889119905 = 12057811198721 + 12057821198722 + 12057831198723

(39)

This completes the proof

Journal of Function Spaces 5

Theorem 23 Under the conditions of Theorem 22 if 119891 and 119892are similarly ordered functions then we have

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 le 12057811198731 + 12057821198732 (40)

where 1205781 and 1205782 are given by (33) and (34)

1198731 = int10

119905120572119904119889119905 = 1120572119904 + 1

1198732 = int10

(1 minus 119905120572)119904 119889119905 = 120573 (119904 + 1 1120572 + 1)

(41)

Proof Integrating inequality (21) completes the proof

Theorem 24 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119902 ge 1 with 120572 isin (0 1] and 119904 isin [0 1]then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(42)

where 1205821 is given by (10)

1205811 = int10

|1 minus 2119905| 119905120572119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (43)

and

1205812 = int10

|1 minus 2119905| (1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (44)

Proof Using Lemma 11 and power mean inequality we have

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816 le 119886119887 (119887 minus 119886)

2sdot int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times (int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816

sdot 100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times ( |1 minus 2119905|(119905119887 + (1 minus 119905) 119886)2 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1 minus 119905120572)119904

sdot 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2

sdot 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902 (45)

This completes the proof

Corollary 25 Under the conditions of Theorem 24 if 119902 = 1then we have100381610038161003816100381610038161003816100381610038161003816

119891 (119886) + 119891 (119887)2 minus 119886119887

119887 minus 119886 int119887119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 (1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816)

(46)

where 1205811 and 1205812 are given by (43) and (44) respectively

Remark 26 If we take 120572 = 119904 = 1 in Theorem 24 theninequality (42) becomes inequality (9) of Theorem 7

If we take 120572 = 1 in Theorem 24 then we have result forharmonically 119904-convex functions in second sense

Corollary 27 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in second sense on [119886 119887] for 119902 gt 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205882 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(47)

where 1205821 is given by (10)

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (48)

and

1205882 = int10

|1 minus 2119905| (1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (49)

respectively

If we take 119904 = 1 in Theorem 24 then we have followingresult for harmonically 119904-convex functions in first sense for120572 = 119904Corollary 28 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in first sense then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 120588lowast 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(50)

6 Journal of Function Spaces

where 1205821 is given by (10) and

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (51)

and

120588lowast = 1205821 minus 1205881 (52)

If we take 119904 = 0 in Theorem 24 we have the followingresult for harmonically 119875 -functions

Corollary 29 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function for 119902 ge 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821 [100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(53)

where 1205821 is given by (10)

Theorem 30 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 with 120572 isin(0 1] 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(54)

where

1205951 = int10

119905120572119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (55)

and

1205952 = int10

(1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (56)

respectively

Proof From Lemma 11 Holderrsquos inequality and the fact that|1198911015840|119902 is harmonically (120572 119904) -convex function on [119886 119887] we have100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0|1

minus 2119905|119901 119889119905)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902

100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1

minus 119905120572)119904 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(57)

where

int10

|1 minus 2119905|119901 119889119905 = 1119901 + 1 (58)

This completes the proof

Remark 31 If we take 120572 = 119904 = 1 in Theorem 30 theninequality (54) becomes inequality (13) of Theorem 8

If we take 120572 = 1 in Theorem 30 then we have result forharmonically 119904-convex functions in second sense

Corollary 32 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in second sense on [119886 119887] for 119901 119902 gt 1 1119901 +1119902 = 1 with 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205931 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205932 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(59)

where

1205931 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (60)

and

1205932 = int10

(1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (119904 + 1 1)1198872119902 21198651 (2119902 119904 + 1 119904 + 2 1 minus 119886

119887 ) (61)

respectively

If we take 119904 = 1 inTheorem 30 thenwe have the followingresult for harmonically 119904-convex functions in first sense for120572 = 119904

Journal of Function Spaces 7

Corollary 33 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in first sense on [119886 119887] for 119901 119902 gt 1 1119901+1119902 = 1 with119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205991 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205992 100381610038161003816100381610038161198911015840 (b)10038161003816100381610038161003816119902)1119902

(62)

where

1205991 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (63)

and

1205992 = int10

1 minus 119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) minus 1205991 (64)

respectively

If 119904 = 0 in Theorem 30 then we have result forharmonically 119875-functionsCorollary 34 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot 1205921119902 (100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(65)

where

120592 = int10

1(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) (66)

4 Some Applications to Special Means

Let us recall the following special means for arbitrary realnumbers 119886 and 119887 (119886 = 119887)(1) The arithmetic mean

119860 fl 119860 (119886 119887) = 119886 + 1198872 (67)

(2) The geometric mean

119866 fl 119866 (119886 119887) = radic119886119887 (68)

(3) The 119901-logarithmic mean

119871119901 fl 119871119901 (119886 119887) = [ 119887119901+1 minus 119886119901+1(119901 + 1) (119887 minus 119886)]

1119901

119901 isin R minus1 0

(69)

Proposition 35 Let 0 lt 119886 lt 119887 119902 gt 1 and 119904 isin (0 1) Then wehave

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 1205821minus11199021 (1205881119886119904 + 1205882119887119904)1119902(70)

where 1205821 1205881 and 1205882 are given by (10) (48) and (49)respectively

Proof The assertion follows from inequality (47) in Corol-lary 27 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)Proposition 36 Let 0 lt 119886 lt 119887 119901 gt 1 119902 = 119901(119901 minus 1) and119904 isin (0 1) Then we have

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 ( 1119901 + 1)1119901 (1205991119886119904 + 1205992119887119904)1119902

(71)

where 1205991 and 1205992 are given by (63) and (64)

Proof The assertion follows from inequality (62) in Corol-lary 33 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)5 Conclusion

In this paper we introduce and investigate a new class ofharmonically convex functions which is called harmoni-cally (120572 119904)-convex functions Some new results of Hermite-Hadamard type for the newly introduced class of harmoni-cally convex functions are established Some applications ofthese results to special means have also been presented Theresults of this paper may stimulate further research for theresearchers working in this field

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

References

[1] GDAndersonMKVamanamurthy andMVuorinen ldquoGen-eralized convexity and inequalitiesrdquo Journal of MathematicalAnalysis and Applications vol 335 no 2 pp 1294ndash1308 2007

8 Journal of Function Spaces

[2] W W Breckner ldquoStetigkeitsaussagen Fur Eine Klasse Verall-gemeinerter Konvexer Funktionen In Topologischen LinearenRaumenrdquo Publications de lrsquoInstitutMathematique vol 23 no 37pp 13ndash20 1978

[3] G Critescu and L Lupsa Non-Connected Convexities andApplications KluwerAcademic Publishers DordrechtHolland2002

[4] V G Mihesan ldquoA generalization of the convexityrdquo in Proceed-ings of the Seminer on Functional Equations Approximation andConvexity Cluj-Napoca Romania 1993

[5] C P Niculescu and L E Persson Convex Functions And TheirApplications Springer New York NY USA 2006

[6] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000

[7] C P Niculescu ldquoConvexity according to meansrdquoMathematicalInequalities amp Applications vol 6 no 4 pp 571ndash579 2003

[8] J Pecaricrsquo F Proschan and Y L TongConvex Functions PartialOrderings and Statistical Applications Academic Press NewYork NY USA 1992

[9] S Varosanec ldquoOn h-convexityrdquo Journal of Mathematical Analy-sis and Applications vol 326 no 1 pp 303ndash311 2007

[10] I Iscan ldquoHermite-Hadamard type inequalities for harmonicallyconvex functionsrdquoHacettepe Journal of Mathematics and Statis-tics vol 43 no 6 pp 935ndash942 2014

[11] I A Baloch and I Iscan ldquoSome Hermite-Hadamard typeinequalities for harmonically (sm)-convex functions in secondsenserdquo 2016 httpsarxivorgabs160408445

[12] I Iscan ldquoHermite-Hadamard type inequalities for harmonically(120572m)-convex functionsrdquoHacettepe Journal of Mathematics andStatistics vol 45 no 2 pp 381ndash390 2016

[13] I Iscan ldquoOstrowski type inequalities for harmonically s-convexfunctionsrdquo Konuralp Journal of Mathematics vol 3 no 1 pp63ndash74 2015

[14] M A Noor K I Noor and M U Awan ldquoSome characteri-zations of harmonically log-convex functionsrdquo Proceedings ofthe Jangjeon Mathematical Society Memoirs of the JangjeonMathematical Society vol 17 no 1 pp 51ndash61 2014

[15] M A Noor K I Noor M U Awan and S Costache ldquoSomeintegral inequalities for harmonically h-convex functionsrdquoldquoPolitehnicardquo University of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 77 no 1 pp 5ndash16 2015

[16] T-Y Zhang A-P Ji and FQi ldquoIntegral inequalities ofHermite-HADamard type for harmonically quasi-convex functionsrdquoProceedings of the JangjeonMathematical SocietyMemoirs of theJangjeon Mathematical Society vol 16 no 3 pp 399ndash407 2013

[17] M W Alomari M Darus and U S Kirmaci ldquoSome inequal-ities of Hermite-Hadamard type for s-convex functionsrdquo ActaMathematica Scientia vol 31 no 4 pp 1643ndash1652 2011

[18] S S Dragomir ldquoHermite-Hadamardrsquos type inequalities forconvex functions of selfadjoint operators in HILbert spacesrdquoLinear Algebra and its Applications vol 436 no 5 pp 1503ndash15152012

[19] S S Dragomir and C E M Pearce Selected Topics on Hermite-Hadamard Inequalities and Applications Victoria UniversityFootscray Australia 2000

[20] S S Dragomir J Pecaric and L E Persson ldquoSome inequalitiesof Hadamard typerdquo Soochow Journal of Mathematics vol 21 no3 pp 335ndash341 1995

[21] H Hudzik and L Maligranda ldquoSome remarks on s-convexfunctionsrdquo Aequationes Mathematicae vol 48 no 1 pp 100ndash111 1994

[22] I Iscan ldquoHermite-Hadamard type inequalities for GA-s-convexfunctionsrdquo Le Matematiche vol 69 no 2 pp 129ndash146 2014

[23] B Y Xi D D Gao and F Qi ldquoIntegral Inequalities ofHermitendashHadamard Type for (120572 s)-Convex And (120572 s m)-Convex Functionsrdquo in HAL Archives p 9 2018 httpshalarchives-ouvertesfrhal-01761678

[24] T-Y Zhang A-P Ji and F Qi ldquoSome inequalities of Hermite-HADamard type for GA-convex functions with applications tomeansrdquo Le Matematiche vol 68 no 1 pp 229ndash239 2013

[25] I Iscan S Numan and K Bekar ldquoHermite-Hadamard andSimpson type inequalities for differentiable harmonically P-convex functionsrdquo British Journal of Mathematics amp ComputerScience vol 4 no 14 pp 1908ndash1920 2014

4 Journal of Function Spaces

Proof Since 119891 is a harmonically (120572 119904)-convex function wehave for all 119909 119910 isin 119868

119891 ( 119909119910119905119910 + (1 minus 119905) 119909) le 119905120572119904119891 (119909) + (1 minus 119905120572)119904 119891 (119910) (25)

which gives

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) le 119905120572119904119891 (119886) + (1 minus 119905120572)119904 119891 (119887)

119891 ( 119886119887119905119886 + (1 minus 119905) 119887) le 119905120572119904119891 (119887) + (1 minus 119905120572)119904 119891 (119886)

(26)

for all 119905 isin [0 1]Adding the above inequalities we have

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) + 119891 ( 119886119887

119905119886 + (1 minus 119905) 119887)le [119891 (119886) + 119891 (119887)] [119905120572119904 + (1 minus 119905120572)119904]

(27)

Integrating the above inequality on [0 1] we obtainint10

119891 ( 119886119887119905119887 + (1 minus 119905) 119886) 119889119905 + int1

0119891 ( 119886119887

119905119886 + (1 minus 119905) 119887) 119889119905

le [119891 (119886) + 119891 (119887)] int10

[119905120572119904 + (1 minus 119905120572)119904] 119889119905(28)

which implies

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909

le 119891 (119886) + 119891 (119887)2 int1

0[119905120572119904 + (1 minus 119905120572)119904] 119889119905

(29)

which is the required result

Remark 19 If we take 120572 = 119904 = 1 in Theorem 18 theninequality (24) becomes the right-hand side of inequality (2)

If we take 120572 = 1 in Theorem 18 we have the followingresult for harmonically 119904-convex functions in second sense

Corollary 20 Let 119891 119868 997888rarr R be harmonically 119904-convexfunction in second sense where 119886 119887 isin 119868 with 119886 lt 119887 and119904 isin [0 1] If 119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909 le 119891 (119886) + 119891 (119887)

119904 + 1 (30)

If we take 119904 = 0 then Theorem 18 collapses to thefollowing result

Corollary 21 Let 119891 119868 997888rarr R be harmonically 119875-functionwhere 119886 119887 isin 119868 with 119886 lt 119887 If 119891 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909 le 119891 (119886) + 119891 (119887) (31)

Our coming result is the Hermite-Hadamard inequalityfor product of two harmonically (120572 119904)-convex functions

Theorem 22 Let 119891 119892 119868 997888rarr R be two harmonically (120572 119904)-convex functions where 119886 119887 isin 119868 with 119886 lt 119887 120572 isin (0 1] 119904 isin[0 1] If 119891119892 isin 119871[119886 119887] then

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 le 12057811198721 + 12057821198722 + 12057831198723 (32)

where

1205781 = 119891 (119886) 119892 (119886) (33)

1205782 = 119891 (119887) 119892 (119887) (34)

1205783 = 119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886) (35)

and

1198721 = int10

1199052120572119904119889119905 = 12120572119904 + 1

1198722 = int10

(1 minus 119905120572)2119904 119889119905 = 120573 (2119904 + 1 1120572 + 1)

1198723 = int10

119905120572119904 (1 minus 119905120572)119904 119889119905

(36)

120573 is Euler Beta function defined by

120573 (119909 119910) = int10

119905119909minus1 (1 minus 119905)119910minus1 119889119905 119909 119910 gt 0 (37)

and 21198651 is hypergeometric function defined by

21198651 (119886 119887 119888 119911)= 1

120573 (119887 119888 minus 119887) int10

119905119887minus1 (1 minus 119905)119888minus119887minus1 (1 minus 119911119905)minus119886 119889119905119888 gt 119887 gt 0 |119911| lt 1

(38)

Proof Let 119891 119892 119868 997888rarr R be harmonically (120572 119904)-convexfunctions then we have

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 = int10

119891 ( 119909119910119905119910 + (1 minus 119905) 119909)

sdot 119892 ( 119909119910119905119910 + (1 minus 119905) 119909) 119889119905 le int1

0[119905120572119904119891 (119886)

+ (1 minus 119905120572)119904 119891 (119887)] [119905120572119904119892 (119886) + (1 minus 119905120572)119904 119892 (119887)] 119889119905= int10

(1199052120572119904119891 (119886) 119892 (119886) + (1 minus 119905120572)2119904 119891 (119887) 119892 (119887)+ 119905120572119904 (1 minus 119905120572)119904 [119891 (119886) 119892 (119887) + 119891 (119887) 119892 (119886)]) 119889119905= 1205781 int

1

01199052120572119904119889119905 + 1205782 int

1

0(1 minus 119905120572)2119904 119889119905 + 1205783 int

1

0119905120572119904 (1

minus 119905120572)119904 119889119905 = 12057811198721 + 12057821198722 + 12057831198723

(39)

This completes the proof

Journal of Function Spaces 5

Theorem 23 Under the conditions of Theorem 22 if 119891 and 119892are similarly ordered functions then we have

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 le 12057811198731 + 12057821198732 (40)

where 1205781 and 1205782 are given by (33) and (34)

1198731 = int10

119905120572119904119889119905 = 1120572119904 + 1

1198732 = int10

(1 minus 119905120572)119904 119889119905 = 120573 (119904 + 1 1120572 + 1)

(41)

Proof Integrating inequality (21) completes the proof

Theorem 24 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119902 ge 1 with 120572 isin (0 1] and 119904 isin [0 1]then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(42)

where 1205821 is given by (10)

1205811 = int10

|1 minus 2119905| 119905120572119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (43)

and

1205812 = int10

|1 minus 2119905| (1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (44)

Proof Using Lemma 11 and power mean inequality we have

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816 le 119886119887 (119887 minus 119886)

2sdot int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times (int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816

sdot 100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times ( |1 minus 2119905|(119905119887 + (1 minus 119905) 119886)2 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1 minus 119905120572)119904

sdot 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2

sdot 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902 (45)

This completes the proof

Corollary 25 Under the conditions of Theorem 24 if 119902 = 1then we have100381610038161003816100381610038161003816100381610038161003816

119891 (119886) + 119891 (119887)2 minus 119886119887

119887 minus 119886 int119887119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 (1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816)

(46)

where 1205811 and 1205812 are given by (43) and (44) respectively

Remark 26 If we take 120572 = 119904 = 1 in Theorem 24 theninequality (42) becomes inequality (9) of Theorem 7

If we take 120572 = 1 in Theorem 24 then we have result forharmonically 119904-convex functions in second sense

Corollary 27 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in second sense on [119886 119887] for 119902 gt 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205882 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(47)

where 1205821 is given by (10)

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (48)

and

1205882 = int10

|1 minus 2119905| (1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (49)

respectively

If we take 119904 = 1 in Theorem 24 then we have followingresult for harmonically 119904-convex functions in first sense for120572 = 119904Corollary 28 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in first sense then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 120588lowast 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(50)

6 Journal of Function Spaces

where 1205821 is given by (10) and

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (51)

and

120588lowast = 1205821 minus 1205881 (52)

If we take 119904 = 0 in Theorem 24 we have the followingresult for harmonically 119875 -functions

Corollary 29 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function for 119902 ge 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821 [100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(53)

where 1205821 is given by (10)

Theorem 30 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 with 120572 isin(0 1] 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(54)

where

1205951 = int10

119905120572119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (55)

and

1205952 = int10

(1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (56)

respectively

Proof From Lemma 11 Holderrsquos inequality and the fact that|1198911015840|119902 is harmonically (120572 119904) -convex function on [119886 119887] we have100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0|1

minus 2119905|119901 119889119905)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902

100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1

minus 119905120572)119904 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(57)

where

int10

|1 minus 2119905|119901 119889119905 = 1119901 + 1 (58)

This completes the proof

Remark 31 If we take 120572 = 119904 = 1 in Theorem 30 theninequality (54) becomes inequality (13) of Theorem 8

If we take 120572 = 1 in Theorem 30 then we have result forharmonically 119904-convex functions in second sense

Corollary 32 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in second sense on [119886 119887] for 119901 119902 gt 1 1119901 +1119902 = 1 with 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205931 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205932 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(59)

where

1205931 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (60)

and

1205932 = int10

(1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (119904 + 1 1)1198872119902 21198651 (2119902 119904 + 1 119904 + 2 1 minus 119886

119887 ) (61)

respectively

If we take 119904 = 1 inTheorem 30 thenwe have the followingresult for harmonically 119904-convex functions in first sense for120572 = 119904

Journal of Function Spaces 7

Corollary 33 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in first sense on [119886 119887] for 119901 119902 gt 1 1119901+1119902 = 1 with119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205991 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205992 100381610038161003816100381610038161198911015840 (b)10038161003816100381610038161003816119902)1119902

(62)

where

1205991 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (63)

and

1205992 = int10

1 minus 119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) minus 1205991 (64)

respectively

If 119904 = 0 in Theorem 30 then we have result forharmonically 119875-functionsCorollary 34 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot 1205921119902 (100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(65)

where

120592 = int10

1(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) (66)

4 Some Applications to Special Means

Let us recall the following special means for arbitrary realnumbers 119886 and 119887 (119886 = 119887)(1) The arithmetic mean

119860 fl 119860 (119886 119887) = 119886 + 1198872 (67)

(2) The geometric mean

119866 fl 119866 (119886 119887) = radic119886119887 (68)

(3) The 119901-logarithmic mean

119871119901 fl 119871119901 (119886 119887) = [ 119887119901+1 minus 119886119901+1(119901 + 1) (119887 minus 119886)]

1119901

119901 isin R minus1 0

(69)

Proposition 35 Let 0 lt 119886 lt 119887 119902 gt 1 and 119904 isin (0 1) Then wehave

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 1205821minus11199021 (1205881119886119904 + 1205882119887119904)1119902(70)

where 1205821 1205881 and 1205882 are given by (10) (48) and (49)respectively

Proof The assertion follows from inequality (47) in Corol-lary 27 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)Proposition 36 Let 0 lt 119886 lt 119887 119901 gt 1 119902 = 119901(119901 minus 1) and119904 isin (0 1) Then we have

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 ( 1119901 + 1)1119901 (1205991119886119904 + 1205992119887119904)1119902

(71)

where 1205991 and 1205992 are given by (63) and (64)

Proof The assertion follows from inequality (62) in Corol-lary 33 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)5 Conclusion

In this paper we introduce and investigate a new class ofharmonically convex functions which is called harmoni-cally (120572 119904)-convex functions Some new results of Hermite-Hadamard type for the newly introduced class of harmoni-cally convex functions are established Some applications ofthese results to special means have also been presented Theresults of this paper may stimulate further research for theresearchers working in this field

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

References

[1] GDAndersonMKVamanamurthy andMVuorinen ldquoGen-eralized convexity and inequalitiesrdquo Journal of MathematicalAnalysis and Applications vol 335 no 2 pp 1294ndash1308 2007

8 Journal of Function Spaces

[2] W W Breckner ldquoStetigkeitsaussagen Fur Eine Klasse Verall-gemeinerter Konvexer Funktionen In Topologischen LinearenRaumenrdquo Publications de lrsquoInstitutMathematique vol 23 no 37pp 13ndash20 1978

[3] G Critescu and L Lupsa Non-Connected Convexities andApplications KluwerAcademic Publishers DordrechtHolland2002

[4] V G Mihesan ldquoA generalization of the convexityrdquo in Proceed-ings of the Seminer on Functional Equations Approximation andConvexity Cluj-Napoca Romania 1993

[5] C P Niculescu and L E Persson Convex Functions And TheirApplications Springer New York NY USA 2006

[6] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000

[7] C P Niculescu ldquoConvexity according to meansrdquoMathematicalInequalities amp Applications vol 6 no 4 pp 571ndash579 2003

[8] J Pecaricrsquo F Proschan and Y L TongConvex Functions PartialOrderings and Statistical Applications Academic Press NewYork NY USA 1992

[9] S Varosanec ldquoOn h-convexityrdquo Journal of Mathematical Analy-sis and Applications vol 326 no 1 pp 303ndash311 2007

[10] I Iscan ldquoHermite-Hadamard type inequalities for harmonicallyconvex functionsrdquoHacettepe Journal of Mathematics and Statis-tics vol 43 no 6 pp 935ndash942 2014

[11] I A Baloch and I Iscan ldquoSome Hermite-Hadamard typeinequalities for harmonically (sm)-convex functions in secondsenserdquo 2016 httpsarxivorgabs160408445

[12] I Iscan ldquoHermite-Hadamard type inequalities for harmonically(120572m)-convex functionsrdquoHacettepe Journal of Mathematics andStatistics vol 45 no 2 pp 381ndash390 2016

[13] I Iscan ldquoOstrowski type inequalities for harmonically s-convexfunctionsrdquo Konuralp Journal of Mathematics vol 3 no 1 pp63ndash74 2015

[14] M A Noor K I Noor and M U Awan ldquoSome characteri-zations of harmonically log-convex functionsrdquo Proceedings ofthe Jangjeon Mathematical Society Memoirs of the JangjeonMathematical Society vol 17 no 1 pp 51ndash61 2014

[15] M A Noor K I Noor M U Awan and S Costache ldquoSomeintegral inequalities for harmonically h-convex functionsrdquoldquoPolitehnicardquo University of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 77 no 1 pp 5ndash16 2015

[16] T-Y Zhang A-P Ji and FQi ldquoIntegral inequalities ofHermite-HADamard type for harmonically quasi-convex functionsrdquoProceedings of the JangjeonMathematical SocietyMemoirs of theJangjeon Mathematical Society vol 16 no 3 pp 399ndash407 2013

[17] M W Alomari M Darus and U S Kirmaci ldquoSome inequal-ities of Hermite-Hadamard type for s-convex functionsrdquo ActaMathematica Scientia vol 31 no 4 pp 1643ndash1652 2011

[18] S S Dragomir ldquoHermite-Hadamardrsquos type inequalities forconvex functions of selfadjoint operators in HILbert spacesrdquoLinear Algebra and its Applications vol 436 no 5 pp 1503ndash15152012

[19] S S Dragomir and C E M Pearce Selected Topics on Hermite-Hadamard Inequalities and Applications Victoria UniversityFootscray Australia 2000

[20] S S Dragomir J Pecaric and L E Persson ldquoSome inequalitiesof Hadamard typerdquo Soochow Journal of Mathematics vol 21 no3 pp 335ndash341 1995

[21] H Hudzik and L Maligranda ldquoSome remarks on s-convexfunctionsrdquo Aequationes Mathematicae vol 48 no 1 pp 100ndash111 1994

[22] I Iscan ldquoHermite-Hadamard type inequalities for GA-s-convexfunctionsrdquo Le Matematiche vol 69 no 2 pp 129ndash146 2014

[23] B Y Xi D D Gao and F Qi ldquoIntegral Inequalities ofHermitendashHadamard Type for (120572 s)-Convex And (120572 s m)-Convex Functionsrdquo in HAL Archives p 9 2018 httpshalarchives-ouvertesfrhal-01761678

[24] T-Y Zhang A-P Ji and F Qi ldquoSome inequalities of Hermite-HADamard type for GA-convex functions with applications tomeansrdquo Le Matematiche vol 68 no 1 pp 229ndash239 2013

[25] I Iscan S Numan and K Bekar ldquoHermite-Hadamard andSimpson type inequalities for differentiable harmonically P-convex functionsrdquo British Journal of Mathematics amp ComputerScience vol 4 no 14 pp 1908ndash1920 2014

Journal of Function Spaces 5

Theorem 23 Under the conditions of Theorem 22 if 119891 and 119892are similarly ordered functions then we have

119886119887119887 minus 119886 int119887

119886(119891 (119909) 119892 (119909)

1199092 ) 119889119909 le 12057811198731 + 12057821198732 (40)

where 1205781 and 1205782 are given by (33) and (34)

1198731 = int10

119905120572119904119889119905 = 1120572119904 + 1

1198732 = int10

(1 minus 119905120572)119904 119889119905 = 120573 (119904 + 1 1120572 + 1)

(41)

Proof Integrating inequality (21) completes the proof

Theorem 24 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119902 ge 1 with 120572 isin (0 1] and 119904 isin [0 1]then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(42)

where 1205821 is given by (10)

1205811 = int10

|1 minus 2119905| 119905120572119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (43)

and

1205812 = int10

|1 minus 2119905| (1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (44)

Proof Using Lemma 11 and power mean inequality we have

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816 le 119886119887 (119887 minus 119886)

2sdot int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times (int10

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816

sdot 100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 (int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816 119889119905)1minus1119902

times ( |1 minus 2119905|(119905119887 + (1 minus 119905) 119886)2 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1 minus 119905120572)119904

sdot 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2

sdot 1205821minus11199021 [1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902 (45)

This completes the proof

Corollary 25 Under the conditions of Theorem 24 if 119902 = 1then we have100381610038161003816100381610038161003816100381610038161003816

119891 (119886) + 119891 (119887)2 minus 119886119887

119887 minus 119886 int119887119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 (1205811 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816 + 1205812 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816)

(46)

where 1205811 and 1205812 are given by (43) and (44) respectively

Remark 26 If we take 120572 = 119904 = 1 in Theorem 24 theninequality (42) becomes inequality (9) of Theorem 7

If we take 120572 = 1 in Theorem 24 then we have result forharmonically 119904-convex functions in second sense

Corollary 27 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in second sense on [119886 119887] for 119902 gt 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205882 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(47)

where 1205821 is given by (10)

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (48)

and

1205882 = int10

|1 minus 2119905| (1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (49)

respectively

If we take 119904 = 1 in Theorem 24 then we have followingresult for harmonically 119904-convex functions in first sense for120572 = 119904Corollary 28 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in first sense then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821minus11199021 [1205881 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 120588lowast 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(50)

6 Journal of Function Spaces

where 1205821 is given by (10) and

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (51)

and

120588lowast = 1205821 minus 1205881 (52)

If we take 119904 = 0 in Theorem 24 we have the followingresult for harmonically 119875 -functions

Corollary 29 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function for 119902 ge 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821 [100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(53)

where 1205821 is given by (10)

Theorem 30 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 with 120572 isin(0 1] 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(54)

where

1205951 = int10

119905120572119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (55)

and

1205952 = int10

(1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (56)

respectively

Proof From Lemma 11 Holderrsquos inequality and the fact that|1198911015840|119902 is harmonically (120572 119904) -convex function on [119886 119887] we have100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0|1

minus 2119905|119901 119889119905)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902

100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1

minus 119905120572)119904 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(57)

where

int10

|1 minus 2119905|119901 119889119905 = 1119901 + 1 (58)

This completes the proof

Remark 31 If we take 120572 = 119904 = 1 in Theorem 30 theninequality (54) becomes inequality (13) of Theorem 8

If we take 120572 = 1 in Theorem 30 then we have result forharmonically 119904-convex functions in second sense

Corollary 32 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in second sense on [119886 119887] for 119901 119902 gt 1 1119901 +1119902 = 1 with 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205931 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205932 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(59)

where

1205931 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (60)

and

1205932 = int10

(1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (119904 + 1 1)1198872119902 21198651 (2119902 119904 + 1 119904 + 2 1 minus 119886

119887 ) (61)

respectively

If we take 119904 = 1 inTheorem 30 thenwe have the followingresult for harmonically 119904-convex functions in first sense for120572 = 119904

Journal of Function Spaces 7

Corollary 33 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in first sense on [119886 119887] for 119901 119902 gt 1 1119901+1119902 = 1 with119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205991 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205992 100381610038161003816100381610038161198911015840 (b)10038161003816100381610038161003816119902)1119902

(62)

where

1205991 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (63)

and

1205992 = int10

1 minus 119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) minus 1205991 (64)

respectively

If 119904 = 0 in Theorem 30 then we have result forharmonically 119875-functionsCorollary 34 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot 1205921119902 (100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(65)

where

120592 = int10

1(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) (66)

4 Some Applications to Special Means

Let us recall the following special means for arbitrary realnumbers 119886 and 119887 (119886 = 119887)(1) The arithmetic mean

119860 fl 119860 (119886 119887) = 119886 + 1198872 (67)

(2) The geometric mean

119866 fl 119866 (119886 119887) = radic119886119887 (68)

(3) The 119901-logarithmic mean

119871119901 fl 119871119901 (119886 119887) = [ 119887119901+1 minus 119886119901+1(119901 + 1) (119887 minus 119886)]

1119901

119901 isin R minus1 0

(69)

Proposition 35 Let 0 lt 119886 lt 119887 119902 gt 1 and 119904 isin (0 1) Then wehave

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 1205821minus11199021 (1205881119886119904 + 1205882119887119904)1119902(70)

where 1205821 1205881 and 1205882 are given by (10) (48) and (49)respectively

Proof The assertion follows from inequality (47) in Corol-lary 27 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)Proposition 36 Let 0 lt 119886 lt 119887 119901 gt 1 119902 = 119901(119901 minus 1) and119904 isin (0 1) Then we have

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 ( 1119901 + 1)1119901 (1205991119886119904 + 1205992119887119904)1119902

(71)

where 1205991 and 1205992 are given by (63) and (64)

Proof The assertion follows from inequality (62) in Corol-lary 33 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)5 Conclusion

In this paper we introduce and investigate a new class ofharmonically convex functions which is called harmoni-cally (120572 119904)-convex functions Some new results of Hermite-Hadamard type for the newly introduced class of harmoni-cally convex functions are established Some applications ofthese results to special means have also been presented Theresults of this paper may stimulate further research for theresearchers working in this field

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

References

[1] GDAndersonMKVamanamurthy andMVuorinen ldquoGen-eralized convexity and inequalitiesrdquo Journal of MathematicalAnalysis and Applications vol 335 no 2 pp 1294ndash1308 2007

8 Journal of Function Spaces

[2] W W Breckner ldquoStetigkeitsaussagen Fur Eine Klasse Verall-gemeinerter Konvexer Funktionen In Topologischen LinearenRaumenrdquo Publications de lrsquoInstitutMathematique vol 23 no 37pp 13ndash20 1978

[3] G Critescu and L Lupsa Non-Connected Convexities andApplications KluwerAcademic Publishers DordrechtHolland2002

[4] V G Mihesan ldquoA generalization of the convexityrdquo in Proceed-ings of the Seminer on Functional Equations Approximation andConvexity Cluj-Napoca Romania 1993

[5] C P Niculescu and L E Persson Convex Functions And TheirApplications Springer New York NY USA 2006

[6] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000

[7] C P Niculescu ldquoConvexity according to meansrdquoMathematicalInequalities amp Applications vol 6 no 4 pp 571ndash579 2003

[8] J Pecaricrsquo F Proschan and Y L TongConvex Functions PartialOrderings and Statistical Applications Academic Press NewYork NY USA 1992

[9] S Varosanec ldquoOn h-convexityrdquo Journal of Mathematical Analy-sis and Applications vol 326 no 1 pp 303ndash311 2007

[10] I Iscan ldquoHermite-Hadamard type inequalities for harmonicallyconvex functionsrdquoHacettepe Journal of Mathematics and Statis-tics vol 43 no 6 pp 935ndash942 2014

[11] I A Baloch and I Iscan ldquoSome Hermite-Hadamard typeinequalities for harmonically (sm)-convex functions in secondsenserdquo 2016 httpsarxivorgabs160408445

[12] I Iscan ldquoHermite-Hadamard type inequalities for harmonically(120572m)-convex functionsrdquoHacettepe Journal of Mathematics andStatistics vol 45 no 2 pp 381ndash390 2016

[13] I Iscan ldquoOstrowski type inequalities for harmonically s-convexfunctionsrdquo Konuralp Journal of Mathematics vol 3 no 1 pp63ndash74 2015

[14] M A Noor K I Noor and M U Awan ldquoSome characteri-zations of harmonically log-convex functionsrdquo Proceedings ofthe Jangjeon Mathematical Society Memoirs of the JangjeonMathematical Society vol 17 no 1 pp 51ndash61 2014

[15] M A Noor K I Noor M U Awan and S Costache ldquoSomeintegral inequalities for harmonically h-convex functionsrdquoldquoPolitehnicardquo University of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 77 no 1 pp 5ndash16 2015

[16] T-Y Zhang A-P Ji and FQi ldquoIntegral inequalities ofHermite-HADamard type for harmonically quasi-convex functionsrdquoProceedings of the JangjeonMathematical SocietyMemoirs of theJangjeon Mathematical Society vol 16 no 3 pp 399ndash407 2013

[17] M W Alomari M Darus and U S Kirmaci ldquoSome inequal-ities of Hermite-Hadamard type for s-convex functionsrdquo ActaMathematica Scientia vol 31 no 4 pp 1643ndash1652 2011

[18] S S Dragomir ldquoHermite-Hadamardrsquos type inequalities forconvex functions of selfadjoint operators in HILbert spacesrdquoLinear Algebra and its Applications vol 436 no 5 pp 1503ndash15152012

[19] S S Dragomir and C E M Pearce Selected Topics on Hermite-Hadamard Inequalities and Applications Victoria UniversityFootscray Australia 2000

[20] S S Dragomir J Pecaric and L E Persson ldquoSome inequalitiesof Hadamard typerdquo Soochow Journal of Mathematics vol 21 no3 pp 335ndash341 1995

[21] H Hudzik and L Maligranda ldquoSome remarks on s-convexfunctionsrdquo Aequationes Mathematicae vol 48 no 1 pp 100ndash111 1994

[22] I Iscan ldquoHermite-Hadamard type inequalities for GA-s-convexfunctionsrdquo Le Matematiche vol 69 no 2 pp 129ndash146 2014

[23] B Y Xi D D Gao and F Qi ldquoIntegral Inequalities ofHermitendashHadamard Type for (120572 s)-Convex And (120572 s m)-Convex Functionsrdquo in HAL Archives p 9 2018 httpshalarchives-ouvertesfrhal-01761678

[24] T-Y Zhang A-P Ji and F Qi ldquoSome inequalities of Hermite-HADamard type for GA-convex functions with applications tomeansrdquo Le Matematiche vol 68 no 1 pp 229ndash239 2013

[25] I Iscan S Numan and K Bekar ldquoHermite-Hadamard andSimpson type inequalities for differentiable harmonically P-convex functionsrdquo British Journal of Mathematics amp ComputerScience vol 4 no 14 pp 1908ndash1920 2014

6 Journal of Function Spaces

where 1205821 is given by (10) and

1205881 = int10

|1 minus 2119905| 119905119904(119905119887 + (1 minus 119905) 119886)2 119889119905 (51)

and

120588lowast = 1205821 minus 1205881 (52)

If we take 119904 = 0 in Theorem 24 we have the followingresult for harmonically 119875 -functions

Corollary 29 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868 with 119886 lt 119887 and 1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function for 119902 ge 1 then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 1205821 [100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902]1119902

(53)

where 1205821 is given by (10)

Theorem 30 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically(120572 119904)-convex on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 with 120572 isin(0 1] 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(54)

where

1205951 = int10

119905120572119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (55)

and

1205952 = int10

(1 minus 119905120572)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 (56)

respectively

Proof From Lemma 11 Holderrsquos inequality and the fact that|1198911015840|119902 is harmonically (120572 119904) -convex function on [119886 119887] we have100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 int1

0

1003816100381610038161003816100381610038161003816100381610038161 minus 2119905

(119905119887 + (1 minus 119905) 119886)2100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816 119889119905

le 119886119887 (119887 minus 119886)2 (int1

0|1

minus 2119905|119901 119889119905)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902

100381610038161003816100381610038161003816100381610038161198911015840 ( 119886119887

119905119887 + (1 minus 119905) 119886)10038161003816100381610038161003816100381610038161003816119902 119889119905)1119902

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (int10

1(119905119887 + (1 minus 119905) 119886)2119902 [119905120572119904 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + (1

minus 119905120572)119904 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902] 119889119905)1119902

= 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901 (1205951 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205952 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(57)

where

int10

|1 minus 2119905|119901 119889119905 = 1119901 + 1 (58)

This completes the proof

Remark 31 If we take 120572 = 119904 = 1 in Theorem 30 theninequality (54) becomes inequality (13) of Theorem 8

If we take 120572 = 1 in Theorem 30 then we have result forharmonically 119904-convex functions in second sense

Corollary 32 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex function in second sense on [119886 119887] for 119901 119902 gt 1 1119901 +1119902 = 1 with 119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205931 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205932 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(59)

where

1205931 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (60)

and

1205932 = int10

(1 minus 119905)119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (119904 + 1 1)1198872119902 21198651 (2119902 119904 + 1 119904 + 2 1 minus 119886

119887 ) (61)

respectively

If we take 119904 = 1 inTheorem 30 thenwe have the followingresult for harmonically 119904-convex functions in first sense for120572 = 119904

Journal of Function Spaces 7

Corollary 33 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in first sense on [119886 119887] for 119901 119902 gt 1 1119901+1119902 = 1 with119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205991 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205992 100381610038161003816100381610038161198911015840 (b)10038161003816100381610038161003816119902)1119902

(62)

where

1205991 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (63)

and

1205992 = int10

1 minus 119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) minus 1205991 (64)

respectively

If 119904 = 0 in Theorem 30 then we have result forharmonically 119875-functionsCorollary 34 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot 1205921119902 (100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(65)

where

120592 = int10

1(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) (66)

4 Some Applications to Special Means

Let us recall the following special means for arbitrary realnumbers 119886 and 119887 (119886 = 119887)(1) The arithmetic mean

119860 fl 119860 (119886 119887) = 119886 + 1198872 (67)

(2) The geometric mean

119866 fl 119866 (119886 119887) = radic119886119887 (68)

(3) The 119901-logarithmic mean

119871119901 fl 119871119901 (119886 119887) = [ 119887119901+1 minus 119886119901+1(119901 + 1) (119887 minus 119886)]

1119901

119901 isin R minus1 0

(69)

Proposition 35 Let 0 lt 119886 lt 119887 119902 gt 1 and 119904 isin (0 1) Then wehave

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 1205821minus11199021 (1205881119886119904 + 1205882119887119904)1119902(70)

where 1205821 1205881 and 1205882 are given by (10) (48) and (49)respectively

Proof The assertion follows from inequality (47) in Corol-lary 27 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)Proposition 36 Let 0 lt 119886 lt 119887 119901 gt 1 119902 = 119901(119901 minus 1) and119904 isin (0 1) Then we have

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 ( 1119901 + 1)1119901 (1205991119886119904 + 1205992119887119904)1119902

(71)

where 1205991 and 1205992 are given by (63) and (64)

Proof The assertion follows from inequality (62) in Corol-lary 33 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)5 Conclusion

In this paper we introduce and investigate a new class ofharmonically convex functions which is called harmoni-cally (120572 119904)-convex functions Some new results of Hermite-Hadamard type for the newly introduced class of harmoni-cally convex functions are established Some applications ofthese results to special means have also been presented Theresults of this paper may stimulate further research for theresearchers working in this field

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

References

[1] GDAndersonMKVamanamurthy andMVuorinen ldquoGen-eralized convexity and inequalitiesrdquo Journal of MathematicalAnalysis and Applications vol 335 no 2 pp 1294ndash1308 2007

8 Journal of Function Spaces

[2] W W Breckner ldquoStetigkeitsaussagen Fur Eine Klasse Verall-gemeinerter Konvexer Funktionen In Topologischen LinearenRaumenrdquo Publications de lrsquoInstitutMathematique vol 23 no 37pp 13ndash20 1978

[3] G Critescu and L Lupsa Non-Connected Convexities andApplications KluwerAcademic Publishers DordrechtHolland2002

[4] V G Mihesan ldquoA generalization of the convexityrdquo in Proceed-ings of the Seminer on Functional Equations Approximation andConvexity Cluj-Napoca Romania 1993

[5] C P Niculescu and L E Persson Convex Functions And TheirApplications Springer New York NY USA 2006

[6] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000

[7] C P Niculescu ldquoConvexity according to meansrdquoMathematicalInequalities amp Applications vol 6 no 4 pp 571ndash579 2003

[8] J Pecaricrsquo F Proschan and Y L TongConvex Functions PartialOrderings and Statistical Applications Academic Press NewYork NY USA 1992

[9] S Varosanec ldquoOn h-convexityrdquo Journal of Mathematical Analy-sis and Applications vol 326 no 1 pp 303ndash311 2007

[10] I Iscan ldquoHermite-Hadamard type inequalities for harmonicallyconvex functionsrdquoHacettepe Journal of Mathematics and Statis-tics vol 43 no 6 pp 935ndash942 2014

[11] I A Baloch and I Iscan ldquoSome Hermite-Hadamard typeinequalities for harmonically (sm)-convex functions in secondsenserdquo 2016 httpsarxivorgabs160408445

[12] I Iscan ldquoHermite-Hadamard type inequalities for harmonically(120572m)-convex functionsrdquoHacettepe Journal of Mathematics andStatistics vol 45 no 2 pp 381ndash390 2016

[13] I Iscan ldquoOstrowski type inequalities for harmonically s-convexfunctionsrdquo Konuralp Journal of Mathematics vol 3 no 1 pp63ndash74 2015

[14] M A Noor K I Noor and M U Awan ldquoSome characteri-zations of harmonically log-convex functionsrdquo Proceedings ofthe Jangjeon Mathematical Society Memoirs of the JangjeonMathematical Society vol 17 no 1 pp 51ndash61 2014

[15] M A Noor K I Noor M U Awan and S Costache ldquoSomeintegral inequalities for harmonically h-convex functionsrdquoldquoPolitehnicardquo University of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 77 no 1 pp 5ndash16 2015

[16] T-Y Zhang A-P Ji and FQi ldquoIntegral inequalities ofHermite-HADamard type for harmonically quasi-convex functionsrdquoProceedings of the JangjeonMathematical SocietyMemoirs of theJangjeon Mathematical Society vol 16 no 3 pp 399ndash407 2013

[17] M W Alomari M Darus and U S Kirmaci ldquoSome inequal-ities of Hermite-Hadamard type for s-convex functionsrdquo ActaMathematica Scientia vol 31 no 4 pp 1643ndash1652 2011

[18] S S Dragomir ldquoHermite-Hadamardrsquos type inequalities forconvex functions of selfadjoint operators in HILbert spacesrdquoLinear Algebra and its Applications vol 436 no 5 pp 1503ndash15152012

[19] S S Dragomir and C E M Pearce Selected Topics on Hermite-Hadamard Inequalities and Applications Victoria UniversityFootscray Australia 2000

[20] S S Dragomir J Pecaric and L E Persson ldquoSome inequalitiesof Hadamard typerdquo Soochow Journal of Mathematics vol 21 no3 pp 335ndash341 1995

[21] H Hudzik and L Maligranda ldquoSome remarks on s-convexfunctionsrdquo Aequationes Mathematicae vol 48 no 1 pp 100ndash111 1994

[22] I Iscan ldquoHermite-Hadamard type inequalities for GA-s-convexfunctionsrdquo Le Matematiche vol 69 no 2 pp 129ndash146 2014

[23] B Y Xi D D Gao and F Qi ldquoIntegral Inequalities ofHermitendashHadamard Type for (120572 s)-Convex And (120572 s m)-Convex Functionsrdquo in HAL Archives p 9 2018 httpshalarchives-ouvertesfrhal-01761678

[24] T-Y Zhang A-P Ji and F Qi ldquoSome inequalities of Hermite-HADamard type for GA-convex functions with applications tomeansrdquo Le Matematiche vol 68 no 1 pp 229ndash239 2013

[25] I Iscan S Numan and K Bekar ldquoHermite-Hadamard andSimpson type inequalities for differentiable harmonically P-convex functionsrdquo British Journal of Mathematics amp ComputerScience vol 4 no 14 pp 1908ndash1920 2014

Journal of Function Spaces 7

Corollary 33 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119904-convex in first sense on [119886 119887] for 119901 119902 gt 1 1119901+1119902 = 1 with119904 isin [0 1] then100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot (1205991 100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 1205992 100381610038161003816100381610038161198911015840 (b)10038161003816100381610038161003816119902)1119902

(62)

where

1205991 = int10

119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905

= 120573 (1 119904 + 1)1198872119902 21198651 (2119902 1 119904 + 2 1 minus 119886

119887 ) (63)

and

1205992 = int10

1 minus 119905119904(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) minus 1205991 (64)

respectively

If 119904 = 0 in Theorem 30 then we have result forharmonically 119875-functionsCorollary 34 Let 119891 119868 997888rarr R be a differentiable function on119868∘ 119886 119887 isin 119868∘ with 119886 lt 119887 and1198911015840 isin 119871[119886 119887] If |1198911015840|119902 is harmonically119875-function on [119886 119887] for 119901 119902 gt 1 1119901 + 1119902 = 1 then

100381610038161003816100381610038161003816100381610038161003816119891 (119886) + 119891 (119887)

2 minus 119886119887119887 minus 119886 int119887

119886

119891 (119909)1199092 119889119909100381610038161003816100381610038161003816100381610038161003816

le 119886119887 (119887 minus 119886)2 ( 1

119901 + 1)1119901

sdot 1205921119902 (100381610038161003816100381610038161198911015840 (119886)10038161003816100381610038161003816119902 + 100381610038161003816100381610038161198911015840 (119887)10038161003816100381610038161003816119902)1119902

(65)

where

120592 = int10

1(119905119887 + (1 minus 119905) 119886)2119902 119889119905 = 1198871minus2119902 minus 1198861minus2119902

(119887 minus 119886) (1 minus 2119902) (66)

4 Some Applications to Special Means

Let us recall the following special means for arbitrary realnumbers 119886 and 119887 (119886 = 119887)(1) The arithmetic mean

119860 fl 119860 (119886 119887) = 119886 + 1198872 (67)

(2) The geometric mean

119866 fl 119866 (119886 119887) = radic119886119887 (68)

(3) The 119901-logarithmic mean

119871119901 fl 119871119901 (119886 119887) = [ 119887119901+1 minus 119886119901+1(119901 + 1) (119887 minus 119886)]

1119901

119901 isin R minus1 0

(69)

Proposition 35 Let 0 lt 119886 lt 119887 119902 gt 1 and 119904 isin (0 1) Then wehave

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 1205821minus11199021 (1205881119886119904 + 1205882119887119904)1119902(70)

where 1205821 1205881 and 1205882 are given by (10) (48) and (49)respectively

Proof The assertion follows from inequality (47) in Corol-lary 27 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)Proposition 36 Let 0 lt 119886 lt 119887 119901 gt 1 119902 = 119901(119901 minus 1) and119904 isin (0 1) Then we have

100381610038161003816100381610038161003816119860 (119886119904119902+1 119887119904119902+1) minus 1198662119871119904119902minus1119904119902minus1

100381610038161003816100381610038161003816le 119886119887 (119887 minus 119886) (119904 + 119902)

2119902 ( 1119901 + 1)1119901 (1205991119886119904 + 1205992119887119904)1119902

(71)

where 1205991 and 1205992 are given by (63) and (64)

Proof The assertion follows from inequality (62) in Corol-lary 33 for (0 infin) 997888rarr R 119891(119909) = 119909119904119902+1(119904119902 + 1)5 Conclusion

In this paper we introduce and investigate a new class ofharmonically convex functions which is called harmoni-cally (120572 119904)-convex functions Some new results of Hermite-Hadamard type for the newly introduced class of harmoni-cally convex functions are established Some applications ofthese results to special means have also been presented Theresults of this paper may stimulate further research for theresearchers working in this field

Data Availability

No data were used to support this study

Conflicts of Interest

The author declares no conflicts of interest

References

[1] GDAndersonMKVamanamurthy andMVuorinen ldquoGen-eralized convexity and inequalitiesrdquo Journal of MathematicalAnalysis and Applications vol 335 no 2 pp 1294ndash1308 2007

8 Journal of Function Spaces

[2] W W Breckner ldquoStetigkeitsaussagen Fur Eine Klasse Verall-gemeinerter Konvexer Funktionen In Topologischen LinearenRaumenrdquo Publications de lrsquoInstitutMathematique vol 23 no 37pp 13ndash20 1978

[3] G Critescu and L Lupsa Non-Connected Convexities andApplications KluwerAcademic Publishers DordrechtHolland2002

[4] V G Mihesan ldquoA generalization of the convexityrdquo in Proceed-ings of the Seminer on Functional Equations Approximation andConvexity Cluj-Napoca Romania 1993

[5] C P Niculescu and L E Persson Convex Functions And TheirApplications Springer New York NY USA 2006

[6] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000

[7] C P Niculescu ldquoConvexity according to meansrdquoMathematicalInequalities amp Applications vol 6 no 4 pp 571ndash579 2003

[8] J Pecaricrsquo F Proschan and Y L TongConvex Functions PartialOrderings and Statistical Applications Academic Press NewYork NY USA 1992

[9] S Varosanec ldquoOn h-convexityrdquo Journal of Mathematical Analy-sis and Applications vol 326 no 1 pp 303ndash311 2007

[10] I Iscan ldquoHermite-Hadamard type inequalities for harmonicallyconvex functionsrdquoHacettepe Journal of Mathematics and Statis-tics vol 43 no 6 pp 935ndash942 2014

[11] I A Baloch and I Iscan ldquoSome Hermite-Hadamard typeinequalities for harmonically (sm)-convex functions in secondsenserdquo 2016 httpsarxivorgabs160408445

[12] I Iscan ldquoHermite-Hadamard type inequalities for harmonically(120572m)-convex functionsrdquoHacettepe Journal of Mathematics andStatistics vol 45 no 2 pp 381ndash390 2016

[13] I Iscan ldquoOstrowski type inequalities for harmonically s-convexfunctionsrdquo Konuralp Journal of Mathematics vol 3 no 1 pp63ndash74 2015

[14] M A Noor K I Noor and M U Awan ldquoSome characteri-zations of harmonically log-convex functionsrdquo Proceedings ofthe Jangjeon Mathematical Society Memoirs of the JangjeonMathematical Society vol 17 no 1 pp 51ndash61 2014

[15] M A Noor K I Noor M U Awan and S Costache ldquoSomeintegral inequalities for harmonically h-convex functionsrdquoldquoPolitehnicardquo University of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 77 no 1 pp 5ndash16 2015

[16] T-Y Zhang A-P Ji and FQi ldquoIntegral inequalities ofHermite-HADamard type for harmonically quasi-convex functionsrdquoProceedings of the JangjeonMathematical SocietyMemoirs of theJangjeon Mathematical Society vol 16 no 3 pp 399ndash407 2013

[17] M W Alomari M Darus and U S Kirmaci ldquoSome inequal-ities of Hermite-Hadamard type for s-convex functionsrdquo ActaMathematica Scientia vol 31 no 4 pp 1643ndash1652 2011

[18] S S Dragomir ldquoHermite-Hadamardrsquos type inequalities forconvex functions of selfadjoint operators in HILbert spacesrdquoLinear Algebra and its Applications vol 436 no 5 pp 1503ndash15152012

[19] S S Dragomir and C E M Pearce Selected Topics on Hermite-Hadamard Inequalities and Applications Victoria UniversityFootscray Australia 2000

[20] S S Dragomir J Pecaric and L E Persson ldquoSome inequalitiesof Hadamard typerdquo Soochow Journal of Mathematics vol 21 no3 pp 335ndash341 1995

[21] H Hudzik and L Maligranda ldquoSome remarks on s-convexfunctionsrdquo Aequationes Mathematicae vol 48 no 1 pp 100ndash111 1994

[22] I Iscan ldquoHermite-Hadamard type inequalities for GA-s-convexfunctionsrdquo Le Matematiche vol 69 no 2 pp 129ndash146 2014

[23] B Y Xi D D Gao and F Qi ldquoIntegral Inequalities ofHermitendashHadamard Type for (120572 s)-Convex And (120572 s m)-Convex Functionsrdquo in HAL Archives p 9 2018 httpshalarchives-ouvertesfrhal-01761678

[24] T-Y Zhang A-P Ji and F Qi ldquoSome inequalities of Hermite-HADamard type for GA-convex functions with applications tomeansrdquo Le Matematiche vol 68 no 1 pp 229ndash239 2013

[25] I Iscan S Numan and K Bekar ldquoHermite-Hadamard andSimpson type inequalities for differentiable harmonically P-convex functionsrdquo British Journal of Mathematics amp ComputerScience vol 4 no 14 pp 1908ndash1920 2014

8 Journal of Function Spaces

[2] W W Breckner ldquoStetigkeitsaussagen Fur Eine Klasse Verall-gemeinerter Konvexer Funktionen In Topologischen LinearenRaumenrdquo Publications de lrsquoInstitutMathematique vol 23 no 37pp 13ndash20 1978

[3] G Critescu and L Lupsa Non-Connected Convexities andApplications KluwerAcademic Publishers DordrechtHolland2002

[4] V G Mihesan ldquoA generalization of the convexityrdquo in Proceed-ings of the Seminer on Functional Equations Approximation andConvexity Cluj-Napoca Romania 1993

[5] C P Niculescu and L E Persson Convex Functions And TheirApplications Springer New York NY USA 2006

[6] C P Niculescu ldquoConvexity according to the geometric meanrdquoMathematical Inequalities amp Applications vol 3 no 2 pp 155ndash167 2000

[7] C P Niculescu ldquoConvexity according to meansrdquoMathematicalInequalities amp Applications vol 6 no 4 pp 571ndash579 2003

[8] J Pecaricrsquo F Proschan and Y L TongConvex Functions PartialOrderings and Statistical Applications Academic Press NewYork NY USA 1992

[9] S Varosanec ldquoOn h-convexityrdquo Journal of Mathematical Analy-sis and Applications vol 326 no 1 pp 303ndash311 2007

[10] I Iscan ldquoHermite-Hadamard type inequalities for harmonicallyconvex functionsrdquoHacettepe Journal of Mathematics and Statis-tics vol 43 no 6 pp 935ndash942 2014

[11] I A Baloch and I Iscan ldquoSome Hermite-Hadamard typeinequalities for harmonically (sm)-convex functions in secondsenserdquo 2016 httpsarxivorgabs160408445

[12] I Iscan ldquoHermite-Hadamard type inequalities for harmonically(120572m)-convex functionsrdquoHacettepe Journal of Mathematics andStatistics vol 45 no 2 pp 381ndash390 2016

[13] I Iscan ldquoOstrowski type inequalities for harmonically s-convexfunctionsrdquo Konuralp Journal of Mathematics vol 3 no 1 pp63ndash74 2015

[14] M A Noor K I Noor and M U Awan ldquoSome characteri-zations of harmonically log-convex functionsrdquo Proceedings ofthe Jangjeon Mathematical Society Memoirs of the JangjeonMathematical Society vol 17 no 1 pp 51ndash61 2014

[15] M A Noor K I Noor M U Awan and S Costache ldquoSomeintegral inequalities for harmonically h-convex functionsrdquoldquoPolitehnicardquo University of Bucharest Scientific Bulletin Series AApplied Mathematics and Physics vol 77 no 1 pp 5ndash16 2015

[16] T-Y Zhang A-P Ji and FQi ldquoIntegral inequalities ofHermite-HADamard type for harmonically quasi-convex functionsrdquoProceedings of the JangjeonMathematical SocietyMemoirs of theJangjeon Mathematical Society vol 16 no 3 pp 399ndash407 2013

[17] M W Alomari M Darus and U S Kirmaci ldquoSome inequal-ities of Hermite-Hadamard type for s-convex functionsrdquo ActaMathematica Scientia vol 31 no 4 pp 1643ndash1652 2011

[18] S S Dragomir ldquoHermite-Hadamardrsquos type inequalities forconvex functions of selfadjoint operators in HILbert spacesrdquoLinear Algebra and its Applications vol 436 no 5 pp 1503ndash15152012

[19] S S Dragomir and C E M Pearce Selected Topics on Hermite-Hadamard Inequalities and Applications Victoria UniversityFootscray Australia 2000

[20] S S Dragomir J Pecaric and L E Persson ldquoSome inequalitiesof Hadamard typerdquo Soochow Journal of Mathematics vol 21 no3 pp 335ndash341 1995

[21] H Hudzik and L Maligranda ldquoSome remarks on s-convexfunctionsrdquo Aequationes Mathematicae vol 48 no 1 pp 100ndash111 1994

[22] I Iscan ldquoHermite-Hadamard type inequalities for GA-s-convexfunctionsrdquo Le Matematiche vol 69 no 2 pp 129ndash146 2014

[23] B Y Xi D D Gao and F Qi ldquoIntegral Inequalities ofHermitendashHadamard Type for (120572 s)-Convex And (120572 s m)-Convex Functionsrdquo in HAL Archives p 9 2018 httpshalarchives-ouvertesfrhal-01761678

[24] T-Y Zhang A-P Ji and F Qi ldquoSome inequalities of Hermite-HADamard type for GA-convex functions with applications tomeansrdquo Le Matematiche vol 68 no 1 pp 229ndash239 2013

[25] I Iscan S Numan and K Bekar ldquoHermite-Hadamard andSimpson type inequalities for differentiable harmonically P-convex functionsrdquo British Journal of Mathematics amp ComputerScience vol 4 no 14 pp 1908ndash1920 2014

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