# Fractional Integral Inequalities of Hermite–Hadamard Type for Differentiable Generalizedh-Convex Functions

Authors: Yingxia Yang; Muhammad Shoaib Saleem; Mamoona Ghafoor; Muhammad Imran Qureshi
Journal: Journal of Mathematics
 (2020)

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## Abstract

In the present paper, some fractional integral inequalities of Hermite–Hadamard type for functions whose derivatives are generalizedh-convex are established. Moreover, several particular cases are also discussed which can be deduced from our results. As special cases, one can obtain several new versions of the results of generalized h-convexity for other various generalizations of convex functions.


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## Body

## 1. Introduction

The theory of inequalities is in process of continuous development state, and inequalities have become very effectual and potent tools for analyzing a large number of problems in different branches of mathematics. In last decades, the theory of inequalities has drawn the attention of most of the researchers, motivated new research directions, and affected several features of mathematical analysis and applications [1–4]. Among a variety of inequalities, some inequalities such as Jensen, Hilbert, Hadamard, Hardy, Poincare, Sobolev, Opial, and Fejér have made a significant influence on numerous branches of mathematics [5–8]. One good strategy for investigating any convexity is first to discuss main inequalities such as Hermite–Hadamard and Fejér and then to derive fractional integral inequalities for this. The fractional inequalities for convex function are also important to calculate different means. So, it is always interesting to develop fractional integral inequalities for some generalized convexity. For recent results concerning fractional Hermite–Hadamard inequalities and for different generalizations of convexity, see [9–11] and references therein.In 2019, some new inequalities based on harmonic log-convex functions have been studied by Baloch et al. in [12]. In the same year, Sarikaya and Alp [13] studied the Hermite–Hadamard–Fejér type integral inequalities for generalized convex functions via local fractional integrals. Kwun et al. [14] studied generalized Riemann–Liouville k-fractional integrals associated with Ostrowski-type inequalities and error bounds of Hadamard inequalities. In [15], Kang et al. studied Hadamard and Fejér–Hadamard inequalities for extended generalized fractional integrals involving special functions. In [15], h,m-convex functions and associated fractional Hadamard and Fejér–Hadamard inequalities via an extended generalized Mittag–Leffler function were studied. Simpson’s type inequalities for strongly s,m−convex functions in the second sense has been studied in [16] by Kermausuor.In the present paper, we aim to study some Hermite–Hadamard-type fractional integral inequalities for functions whose derivatives are generalizedh-convex. Our results can be considered as generalization of many existing results as many existing results can be obtained directly from our results.The paper is organized as follows. In Section2, we give some basic definitions and known results that will help to prove our main results. In Section 3, we will develop Fejér and Hermite–Hadamard-type inequalities for generalized h-convex function. The Section 4 is devoted for fractional integral inequalities for generalized h-convex function. The results are concluded in Section 5.


## 2. Preliminaries

Throughout this research, assumeI⊆ℝ be an interval. Also, η:A×A⟶B be a bifunction, where A,B⊆ℝ. Before turning to main results, we first review the basic definitions.Definition 1.
(see [17]). A function f:I⟶ℝ is called η-convex function if(1)fαa1+1−αa2≤fa2+αηfa1,fa2,for all a1,a2∈I and α∈0,1.
This definition was introduced in [17] and motivated by works carried out in [18, 19]; Delavar and Dragomir proved few basic results of η-convex functions.Definition 2.
(see [20]). Let h:J⊆ℝ⟶ℝ be a nonnegative function. A function f:I⟶ℝ is known as modified h-convex if(2)fαa1+1−αa2≤hαfa1+1−hαfa2,for all a1,a2∈I and α∈0,1.
We now define certain convexity which is unification of generalized and modifiedh-convexity.Definition 3.
(see [21]). Let h be a function defined as above. A function f:I⟶ℝ is called generalized h-convex function if(3)fαa1+1−αa2≤fa2+hαηfa1,fa2,for all a1,a2∈I and α∈0,1.Remark 1.
It is worth pointing out that the Definition3 can ofcourse be reduced to(1)
η-convexity for hα=α(2)
ηs-convexity for hα=αs(3)
Modifiedh-convex function when we take ηa,b=a−b(4)
Convex function when we takehα=α and ηa,b=a−b
A convex functionf:I⟶ℝ, a1,a2∈I with a1<a2 satisfies the following inequality:(4)fa1+a22≤1a2−a1∫a1a2fxdx≤fa1+fa22,known as Hermite–Hadamard’s inequality for convex function.
In [21], Fejér generalized (4) as below.
Assume a convex functionf:a1,a2⟶ℝ and a function g:a1,a2⟶ℝ which is nonnegative, integrable, and symmetric with respect to a1+a2/2; then,(5)fa1+a22∫a1a2gxdx≤∫a1a2fxgxdx≤fa1+fa22∫a1a2gxdx.
Furthermore, Pachpatte developed the following inequalities for product of convex functions, see [22].Theorem 1.
Letf,g:a1,a2⟶0,∞ be convex functions on a1,a2⊂ℝ and a1<a2. Then,(6)1a2−a1∫a1a2fxgxdx≤13Ma1,a2+16Na1,a2,2fa1+a22ga1+a22≤1a2−a1∫a1a2fxgxdx+16Ma1,a2+13Na1,a2,where(7)Ma1,a2=fa1ga1+fa2ga2,Na1,a2=fa1ga2+fa2ga1.
For establishing new fractional integral inequalities, the following Lemmas are required.Lemma 1.
(see [23]). Assume a function f:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2. If f′∈La1,a2, ζ, ν∈ℝ, and α∈0,1, then(8)ζfa1+νfa22+2−ζ−ν2fa1+a22−1a2−a1∫a1a2fxdx=a2−a14∫011−ζ−tf′ta1+1−ta1+a22+ν−tf′ta1+a22+1−ta2dα.Lemma 2.
(see [23]). For s>0 and 0≤ϵ≤1, one has(9)∫01ϵ−tsdt=ϵs+1+1−ϵs+1s+1,∫01tϵ−tsdt=ϵs+2+s+1+ϵ1−ϵs+1s+1s+2.


## 3. Hermite–Hadamard and Fejér-Type Inequalities

Here, we present our main results.Theorem 2.
Assume a generalizedh-convex function f:I⟶ℝ, a1, a2∈I with a1<a2 and f∈La1,a2. Then,(10)fa1+a22−h1/2a2−a1∫a1a2ηfa1+a2−x,fxdx≤1a2−a1∫a1a2fxdx≤fb+ηfa1,fa2∫01hαdα.Proof.
Sincef is a generalized h-convex function, so(11)fαx+1−αy≤fy+hαηfx,fy.
Substitutingx=ta1+1−ta2, y=1−ta1+ta2, and α=1/2 in (11), we obtain(12)fa1+a22≤f1−ta1+ta2+h12ηfta1+1−ta2,f1−ta1+ta2.
By integrating the above inequality, we have(13)fa1+a22≤∫01f1−ta1+ta2dt+h12∫01ηfta1+1−ta2,f1−ta1+ta2dt≤1a2−a1∫a1a2fxdx+h1/2a2−a1∫a1a2ηfa1+a2−x,fxdx.
Now, finally upon recalling (11) with x=a1 and y=a2 and integrating over 0,1, we have(14)1a2−a1∫a1a2fxdx≤fa2+ηfa1,fa2∫01hαdα.
Of course (11) and (14) yields (10).Remark 2.
By puttingηa,b=a−b and hα=α in (10), we get classical Hermite–Hadamard inequality (4) for convex function.Theorem 3.
Assume two nonnegative generalizedh-convex functions f and g, a1, a2∈I and a1<a2 such that fg∈L1a1,a2, then(15)1a2−a1∫a1a2fxgxdx≤fa2ga2+M′a1,a2∫01htdt+N′a1,a2∫01h2tdt,where(16)M′a1,a2=fa2ηga1,ga2+ga2ηfa1,fa2,(17)N′a1,a2=ηfa1,fa2ηga1,ga2.Proof.
First, we note(18)fta1+1−ta2≤fa2+htηfa1,fa2,gta1+1−ta2≤ga2+htηfa1,fa2,for all t∈a1,a2. By using nonnegativity of f and g, we obtain(19)fta1+1−ta2gta1+1−ta2≤fa2ga2+htfa2ηga1,ga2+htga2ηfa1,fa2+h2tηfa1,fa2ηga1,ga2.
On integrating the above inequality over0,1, we obtain(20)∫01fta1+1−ta2gta1+1−ta2dt≤fa2ga2+fa2ηga1,ga2+ga2ηfa1,fa2∫01htdt+ηfa1,fa2ηga1,ga2∫01h2tdt.
The above yields (15) for M′ and N′ as in (16) and (17).Remark 3.
If we takeηa,b=a−b and ht=t in Theorem 3, then (15) reduces to (14).Theorem 4.
Assume a generalizedh-convex function f:I⟶ℝ, a1, a2∈I with a1<a2 and f∈L1a1,a2 and a function g:a1,a2⟶ℝ which is nonnegative, integrable, and symmetric, with respect to a1+a2/2; then,(21)∫a1a2fxgxdx≤fa2+12ηfa1,fa2∫a1a2ha2−xa2−a1+hx−a1a2−a1gxdx.Proof.
Sincef is a generalized h-convex function and g is nonnegative, integrable, and symmetric, with respect to a1+a2/2, we find that(22)∫a1a2fugudu=12∫a1a2fxgxdx+∫a1a2fa1+a2−xga1+a2−xdx=12∫a1a2fx+fa1+a2−ugxdx=12∫a1a2fa2−xa2−a1a1+x−a1a2−a1b+fx−a1a2−a1a1+a2−xa2−a1a2gxdx≤12∫a1a2fa2+ha2−xa2−a1ηfa1,fa2+fa2+hx−a1a2−a1ηfa1,fa2gxdx≤fa2+12ηfa1,fa2∫a1a2ha2−xa2−a+hx−a1a2−a1gxdx.Remark 4.
In Theorem4, if(1)
ηa,b=a−b, h(t) = t, and gx=1, then (21) becomes second inequality in (3)(2)
ηa,b=a−b and h(t) = t; then, (21) becomes second inequality in (4).


## 4. Fractional Integral Inequalities

Theorem 5.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2, 0≤ζ, ν≤1, and f′∈La1,a2. If for q≥1, f′q is generalized h-convex on a1,a2, then(23)ζfa1+νfa22+2−ζ−ν2fa+b2−1a2−a1∫a1a2fxdx≤a2−a1821/q1−2ζ+2ζ21−1/q12f′a2q1−2ζ+2ζ2+ηf′a1q,f′a2q∫01h1+t21−ζ−tdt1/q+1−2ν+2ν21−1/q12f′a2q1−2ν+2ν2+ηf′a1q,f′a2q∫01ht2ν−tdt1/q.Proof.
Forq>1, by using Lemma 3, generalized h- convexity of f′xq and the noted Holder’s integral inequality, we obtain(24)ζfa1+νfa22+2−ζ−ν2fa1+a22−1a2−a1∫a1a2fxdx≤a2−a14∫011−ζ−tf′ta1+1−ta1+a22dt+∫01ν−tf′ta1+a22+1−ta2dt≤a2−a14∫011−ζ−tdt1−1/q∫011−ζ−tf′a2q+h1+t2ηf′a1q,f′a2qdt1/q+∫01ν−tdt1−1/q×∫01ν−tf′a2q+ht2ηf′a1q,f′a2qdt1/q.
By using Lemma2, a direct calculation yields(25)∫011−ζ−tf′a2q+h1+t2ηf′a1q,f′a2qdt=f′a2q∫011−ζ−tdt+ηf′a1q,f′a2q×∫01h1+t21−ζ−tdt=121−2ζ+2ζ2f′a2q+ηf′a1q,f′a2q×∫01h1+t21−ζ−tdt,∫01ν−tf′a2q+ht2ηf′a1q,f′a2qdt=f′a2q∫01ν−tdt+ηf′a1q,f′a2q∫01ht2ν−tdt=121−2ν+2ν2f′a2q+ηf′a1q,f′a2q∫01ht2ν−tdt.
On putting the above two inequalities in (24) and using Lemma 2, we obtain inequality (23) for q>1.
Forq=1, by using Lemmas 3 and 2, we obtain(26)ζfa1+νfa22+2−ζ−ν2fa1+a22−1a2−a1∫a1a2fxdx≤a2−a14∫011−ζ−tf′a2+h1+t2ηf′a1,f′a2dt+∫01ν−tf′a2+ht2ηf′a1,f′a2dt=a2−a14121−2ζ+2ζ2f′a2+ηf′a1,f′a2×∫01h1+t21−ζ−tdt+121−2ν+2ν2f′a2+ηf′a1,f′a2∫01ht2ν−tdt.Remark 5.
By puttingηa,b=a−b and ht=t in Theorem 5, we obtain inequality (10) in [23].
We can derive the following corollary by takingζ=ν in Theorem 5.Corollary 1.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2, 0≤ζ≤1, and f′∈La1,a2. If for q≥1, f′q is generalized h-convex on a1,a2, then(27)ζ2fa1+fa2+1−ζfa1+a22−1a2−a1∫a1a2fxdx≤a2−a1821/q1−2ζ+2ζ21−1/q121−2ζ+2ζ2f′a2q+ηf′a1q,f′a2q∫01h1+t21−ζ−tdt1/q+121−2ζ+2ζ2f′a2q+ηf′a1q,f′a2q×∫01ht2ζ−tdt1/q.
If takingζ=ν=1/2,2/3,1/3, respectively, in Theorem 5, we can obtain the following corollaries.Corollary 2.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2 and f′∈La1,a2. If f′q for q≥1 is generalized h-convex on a1,a2, then(28)12fa1+fa22+fa1+a22−1a2−a1∫a1a2fxdx≤a2−a11622/q14f′a2q+ηf′a1q,f′a2q∫01h1+t212−tdt1/q+14f′a2q+ηf′a1q,f′a2q∫01ht212−tdt1/q,13fa1+fa2+fa+b2−1a2−a1∫a1a2fxdx≤5a2−a1721851/q518f′a2q+ηf′a1q,f′a2q∫01h1+t213−tdt1/q+518f′a2q+ηf′a1q,f′a2q∫01h1+t223−tdt1/q,16fa1+fa2+4fa1+a22−1a2−a1∫a1a2fxdx≤5a2−a1721851/q518f′a2q+ηf′a1q,f′a2q∫01h1+t223−tdt1/q+518f′a2q+ηf′a1q,f′a2q∫01ht213−tdt1/q.
If we putq=1 in Corollary 2, then we have the following inequality.Corollary 3.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2b and f′∈La1,a2. If f′ is generalized h-convex on a1,a2, then(29)12fa1+fa22+fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1412f′a2+ηf′a1,f′a2∫01h1+t2+ht212−tdt,13fa1+fa2+fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1459f′a2+ηf′a1,f′a2∫01h1+t213−tdt+∫01ht223−t,16fa1+fa2+4fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1459f′a2+ηf′a1,f′a2∫01h1+t223−tdt+∫01ht213−t.Remark 6.
By settingηa,b=a−b and ht=t, Corollaries 1–1 reduce to inequalities (15)–(17) in [23].Theorem 6.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2, 0≤ζ, ν≤1, and f′∈La1,a2. If f′q for q≥1 is generalized h-convex on a1,a2, then(30)ζfa1+νfa22+2−ζ−ν2fa1+a22−1a2−a1∫a1a2fxdx≤a2−a141−ζq+1+ζq+1q+1f′a2q+ηf′a1q,f′a2q×∫01h1+t21−ζ−tqdt1/q+1−νq+1+νq+1q+1f′a2q+ηf′a1q,f′a2q∫01ht2ν−tqdt1/q.Proof.
Forq>1, by using Lemma 3, ηh-convexity of f′q, and Holder’s integral inequality, we obtain(31)ζfa1+νfa22+2−ζ−ν2fa1+a22−1a2−a1∫a1a2fxdx≤a2−a14∫011−ζ−tf′ta1+1−ta1+a22dt+∫01ν−tf′ta1+a22+1−tbdt≤a2−a14∫01dt1−1/q∫011−ζ−tqf′a2q+h1+t2×ηf′a1q,f′a2qdt1/q+∫01dt1−1/q×∫01μ−tqf′a2q+ht2ηf′a1q,f′a2dt1/q≤a2−a14∫01dt1−1/q∫011−ζ−tqf′a2q+h1+t2×ηf′a1q,f′a2qdt1/q+∫01dt1−1/q×∫01ν−tqf′a2q+ht2ηf′a1q,f′a2dt1/q≤a2−a14∫011−ζ−tqf′a2q+h1+t2×ηf′a1q,f′a2qdt1/q+∫01ν−tq×f′a2q+ht2ηf′a1q,f′a2dt1/q.
By Lemma2, we obtain(32)∫011−ζ−tqf′a2q+h1+t2ηf′a1q,f′a2qdt=f′a2q∫011−ζ−tqdt+ηf′a1q,f′a2q×∫01h1+t21−ζ−tqdt=f′a2q1−ζq+1+ζq+1q+1+ηf′a1q,f′a2q×∫01h1+t21−ζ−tqdt,∫01ν−tqf′a2q+ht2ηf′a1q,f′a2qdt=f′a2q∫01ν−tqdt+ηf′a1q,f′a2q∫01ht2ν−tqdt=f′a2qνq+1+1−νq+1q+1+ηf′a1q,f′a2q×∫01ht2ν−tqdt.
On putting the above two inequalities in (31), we obtain inequality (30) for q>1.
We can obtain the result forq=1 by following the proof of (26).Remark 7.
By takingηa,b=a−b and ht=t in Theorem 6, we obtain inequality (21) of [23].
Similar to the corollaries of Theorem5, the following corollaries of Theorem 6 can be obtained.Corollary 4.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2, 0≤ζ≤1, and f′∈La1,a2. If f′q for q≥1 is generalized h-convex on a1,a2, then(33)ζ2fa1+fa2+1−ζfa1+a22−1a2−a1∫a1a2fxdx≤a2−a141−ζq+1+ζq+1q+1f′a2q+ηf′a1q,f′a2q×∫01h1+t21−ζ−tqdt1/q+1−ζq+1+ζq+1q+1f′a2q+ηf′a1q,f′a2q∫01ht2ζ−tqdt1/q.Corollary 5.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2 and f′∈La1,a2. If f′q for q≥1 is generalized h-convex on a1,a2, then(34)12fa1+fa22+fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1421/2q+1q+1f′a2q+ηf′a1q,f′a2q∫01h1+t212−tqdt1/q+21/2q+1q+1f′a2q+ηf′a1q,f′bq∫01ht212−tqdt1/q,16fa1+fa2+4fa1+a22−1a2−a1∫a1a2fxdx≤a2−a142/3q+1+1/3q+1q+1f′a2q+ηf′a1q,f′a2q×∫01h1+t223−tqdt1/q+2/3q+1+1/3q+1q+1f′a2q+ηf′a1q,f′a2q∫01ht213−tqdt1/q.
If we putq=1 in Corollary 5, then we have the following corollary.Corollary 6.
Assume a functionf:I⟶ℝ, differentiable on I°, a1, a2∈I with a1<a2, and f′∈La1,a2. If f′ is generalized h-convex on a1,a2, then(35)12fa1+fa22+fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1414f′a2+ηf′a1,f′a2∫01h1+t212−tdt+14f′a2+ηf′a1,f′b∫01ht212−tdt,16fa1+fa2+4fa1+a22−1a2−a1∫a1a2fxdx≤a2−a14518f′a2q+ηf′a1,f′a2∫01h1+t223−tdt+518f′a2+ηf′a1,f′a2∫01ht213−tdt.Remark 8.
Corollaries 4–6 reduce to inequalities3.11,3.12, and3.13 of [23] by setting ηa,b=a−b and ht=t.
The following result is required for the derivation of next theorems.Lemma 3.
(see [24]). Assume a mapping f:I⊂0,∞⟶ℝ, differentiable on I° where a1, a2∈I with a1<a2. If f′′∈La1,a2, then(36)1a2−a1∫a1a2fxdx−fa1+a22=a2−a1216∫01t2f′′ta1+a22+1−ta1dt+∫01t−12f′′ta2+1−ta1+a22dt.Theorem 7.
Assume a mappingf:I⊂0,∞⟶ℝ, differentiable on I° such that f′′∈La1,a2, where a1,a2∈I with a1<a2. If f′′ is generalized h-convex on a1,a2, then(37)fa1+a22−1a2−a1∫a1a2fxdx≤a2−a121613f′′a1+f′′a1+a22+ηf′′a1+a22,f′′a1×∫01t2htdt+ηf′′a2,f′′a1+a22∫01t−12htdt.Proof.
By using Lemma3 and generalized h-convexity of f′′, we obtain(38)fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1216∫01t2f′′ta1+a22+1−tadt+∫01t−12f′′ta2+1−ta1+a22dt≤a2−a1216∫01t2f′′a1+htηf′′a1+a22,f′′a1dt+∫01t−12f′′a1+a22+htηf′′a2,f′′a1+a22dt=a2−a121613f′′a1+∫01t2htηf′′a1+a22,f′′a1dt+13f′′a1+a22+∫01t−12htηf′′a2,f′′a1+a22=a2−a121613f′′a1+f′′a1+a22+ηf′′a1+a22,f′′a1×∫01t2htdt+ηf′′a2,f′′a1+a22∫01t−12htdt.Remark 9.
By takingηa,b=a−b and ht=t in Theorem 7, we obtain inequality (9) in [24].Theorem 8.
Assume a mappingf:I⊂0,∞⟶ℝ differentiable on I° such that f′′∈La1,a2, where a1,a2∈I with a1<a2. If f′′q, q≥1 is generalized h-convex on a1,a2, then the following inequality holds:(39)fa1+a22−1a2−a1∫a1a2fxdx≤a2−a216131/p13f′′a1q+ηf′′a1+a22q,f′′a1q×∫01t2htdt1/q+13f′′a2q+ηf′′a2q,f′′a1+a22q×∫01t−12htdt1/q.Proof.
Letp≥1. By using Lemma 3 and power mean inequality, we obtain(40)fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1216∫01t2f′′ta1+a22+1−ta1dt+∫01t−12f′′ta2+1−ta1+a22dt≤a2−a1216∫01t2dt1/p∫01t2f′′ta1+a22+1−taqdt1/q+b−a216∫01t−12dt1/p∫01t−12f′′ta2+1−ta1+a22qdt1/q.
By using generalizedh-convexity of f′′q, we obtain(41)∫01t2f′′ta1+a22+1−ta1qdt≤13f′′a1q+ηf′′a1+a22q,f′′a1q∫01t2htdt,∫01t−12f′′ta2+1−ta1+a22qdt≤13f′′a1+a22q+ηf′′a2q,f′′a1+a22q∫01t−12htdt.
Therefore, we have(42)fa1+a22−1a2−a1∫a1a2fxdx≤a2−a1216131/p13f′′a1q+ηf′′a1+a22q,f′′a1q∫01t2htdt1/q+13f′′a1+a22q+ηf′′a2q,f′′a1+a22q∫01t−12htdt1/q.Remark 10.
By takingηa,b=a−b in Theorem 8, we obtain inequality (2.11) in [24].


## 5. Conclusions

Convexity has many applications in applied sciences as well as in pure mathematics. In the present report, we studiedh-convex functions. We proved some Hermite–Hadamard-type fractional integral inequalities for functions whose derivatives are generalized h-convex. Moreover, we also gave several particular cases, which can be deduced from our results. One can obtain several new versions of the results of generalized h-convexity for other various generalizations of convex functions as a special case of our results, and our results can be considered as extension of many existing results.



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