# Generalized Fractional Integral Inequalities for MT-Non-Convex andpq-Convex Functions

Authors: Wei Wang; Absar Ul Haq; Muhammad Shoaib Saleem; Muhammad Sajid Zahoor
Journal: Journal of Function Spaces
 (2022)

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## Abstract

Fractional integral inequalities have a wide range of applications in pure and applied mathematics. In the present research, we establish generalized fractional integral inequalities for MT-non-convex functions andpq-convex functions. Our results extended many inequalities already existing in the literature.


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## Body

## 1. Introduction

The study of convex analysis is not modern part of mathematics, but some ancient mathematicians also used the interesting geometry of convex functions and convex sets. However, the subject of convex analysis was started in the mid of 20th century. Many remarkable facts and generalization of convex analysis have been obtained quite recently.Convex analysis is one of the appealing subjects for the researchers of geometry and analysis. The interesting geometric, differentiability, and other facilitating properties of convex functions make it distinct from other subjects. Moreover, the convex function and convex set have diverse applications in mathematical physics, technology, economics, and optimization theory.In the last decades, the connection of convexity got stronger due to rapid development in fractional calculus. Therefore, nowadays, it is appreciable to seek new fractional integral inequalities. Simply, we can say that convexity plays a concrete role in fractional integral inequalities and symmetry theory because of its interesting geometric features.There are many well-known integral inequalities related to convex functions, like Jensen’s inequality [1, 2], Opial-type inequality [3], Simpson inequality [4], Ostrowski inequality [5–7], Hermite–Hadamard inequality [8, 9], Olsen integral inequality [10], Fejér-type inequality [1, 11], Hardy inequality [12], and so on. One of the remarkable inequalities for convex function is the Hermite–Hadamard-type inequality.(1)φu+v2≤1v−u∫uvφηdη≤φu+φv2,where φ:I⊆ℝ⟶ℝ is a convex function on the interval I of real numbers and u,v∈I with u<v. Note that several integral inequalities can be obtained from equation (1). There are several versions of inequality equation (1) in the literature, for example, inequalities of Hermite–Hadamard type for functions whose derivatives’ absolute values are quasi-convex are presented in [8]. In [13], the authors established Hermite–Hadamard-type inequalities for p-convex functions via fractional integrals. In [14], the generalized Hermite–Hadamard inequalities are presented. A new version of the Hermite–Hadamard inequality for Riemann–Liouville fractional integrals is presented in [9]. The MT-convex functions via classical and Riemann–Liouville fractional integrals are studied in [15] by Park. He also established Hermite–Hadamard inequality for MT-convex functions. Mohammad in [16] established Hermite–Hadamard-type inequalities on differentiable coordinates for the same class of functions as studied by Park in [15], and Liua et al. [17] developed this inequality for the same class of functions for classical integrals and fractional integrals. For more details, we refer the readers to [18–20].First of all, we recall few definitions.Definition 1 (p-convex set) [21].
I is said to be p-convex set, if ηup+1−ηvp1/p∈I for all u,v∈I and η∈[0, 1].Definition 2 (p-convex) [13, 22].
A functionφ: I⟶ℝ is said to be p-convex function, if(2)φηup+1−ηvp1/p≤ηφu+1−ηφv,for all u, v, ∈I and η∈[0, 1].Definition 3 (MT-convex) [5, 15–17].
A functionφ:I⊂ℝ⟶ℝ is said to be MT-convex on I, if(3)φηu+1−ηv≤η21−ηφu+1−η2ηφvholds for all u, v, ∈I and η∈ [0, 1].
Now we are ready to extend the form of convexities.Definition 4 (MT-non-convex).
A functionφ: I⊂ℝ⟶ℝ is said to be MT-non-convex. Let I be p-convex set, if(4)ηup+1−ηvp1/p≤η21−ηφu+1−η2ηφvholds for all u, v, ∈I and η∈ [0, 1].Remark 1.
In the above definition, forp = 1, we get MT-convex function, and for p = −1, we get harmonically MT-convex function.Definition 5 ((p-q) convex).
A functionφ: I⊂ℝ⟶ℝ is said to be pq-convex. Let I be p-convex set if(5)φηup+1−ηvp1/p≤ηφuq+1−ηφvq1/pholds for all u,v∈I and η∈[0, 1].Definition 6 (Riemann–Liouville fractional integral) [9].
Letφ∈L1u,v and γ> 0. The right side and left side Riemann–Liouville fractional integrals are initiated by(6)Ju+γφs=1Γγ∫uss−ηγ−1φηdη,s>u,Jv−γφs=1Γγ∫svη−sγ−1φηdη,s<v,respectively. For details, we refer the readers to [6, 23].
Now, we define some special functions:(1)
Gamma function:(7)Γz=∫0∞e−ηηz−1dη,z>0.(2)
Beta function:(8)βu,v=ΓuΓvΓu+v=∫01ηu−11−ηv−1dηu,v>0(see [24–27]).(3)
The hypergeometric function [18]:(9)2F1u,v,w,x=1βv,w−v∫01ηv−11−ηw−v−11−dη−udη,w>v>0,x<1.
In [28], Raina introduced a function initiated by(10)Fϱ,λςu=Fϱ,λς0,ς1,…u=∑k=0∞ςkΓϱk+λuk,where ϱ,λ∈ℝ+;u<ℝ and the coefficients ςk∈ℝ+. By using equation (10), Raina and Agarwal [28, 29] initiated the following left and right-side fractional integral operators:(11)Jϱ,λ,u+;ωςφz=∫uzz−ηλ−1Fϱ,λςωz−ηϱφηdη,z>u,Jϱ,λ,v−;ωςφz=∫vzη−zλ−1Fϱ,λςωη−zϱφηdη,z<v,where ω∈ℝ and λ, ϱ∈ℝ+.
This paper is organized as follows. In Section2, we will derive generalized fractional integral inequalities for MT-non-convex function. However, the last section is dedicated to establish generalized fractional integral inequalities for (p-q) convex function.


## 2. Fractional Integral Inequalities for MT-Non-Convex Function

The following lemma is useful to derive our main results.Lemma 1 (see [22]).
Letλ, ∈ℝ+φ:I⊆ℝ+⟶ℝ+ be a differentiable mapping on Iou, v, ∈I such that u < v. If φ′∈L1u,v, p > 0, then we obtain(12)φu+φv2−12Fϱ,λ+1ςωvp−upϱvp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup=vp−up2pFϱ,λ+1ςvp−upϱ∫011−ηλFϱ,λ+1ςωvp−upϱ1−ηϱηup+1−ηvp1/p−1φ′ηup+1−ηvp1/pdη+∫01ηλFϱ,λ+1ςωvp−upϱηϱηup+1−ηvp1/p−1φ′ηup+1−ηvp1/pdη,where ϕc=c1/p.Theorem 1.
Letλ, ∈, ℝ+φ: I⊆ℝ+⟶ℝ+ be a differentiable mapping on Iou,v∈Io such that u<, v. If φ′ is MT-non-convex on u,v, p > 0, then we obtain(13)φu+φv2−12Fϱ,λ+1ςωvp−upϱvp−upλJϱ,λ,up+;ωσφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤Fϱ,λ+1C1ωvp−upϱ+Fϱ,λ+1C3ωvp−upϱφ′u+Fϱ,λ+1C2ωvp−upϱ+Fϱ,λ+1C4ωvp−upϱφ′v,where(14)C1=ςkv1−pϱk+λ−1/23ϱk+λ+1F211−1p,32;ϱk+λ+2,1−upvp,C2=ςkv1−pϱk+λ+1/2ϱk+λ+1F211−1p,12;ϱk+λ+2,1−upvp,C3=ςkv1−p5/4ϱk+λ+3/2ϱk+λ+1F211−1p,ϱk+λ+32;ϱk+λ+2,1−upvp,C4=ςkv1−p14ϱk+λ+1/2ϱk+λ+1F211−1p,ϱk+λ+12;ϱk+λ+2,1−upvp.Proof.
Employing Lemma1 and definition of MT-non-convexity of φ′, we obtain(15)φu+φv2−12Fϱ,λ+1ςωvp−upϱvp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup=vp−up2pFϱ,λ+1ςωvp−upϱ∫011−ηλFϱ,λ+1ςωvp−upϱ1−ηϱηup+1−ηvp1p−1φ′ηup+1−ηvp1pdη+∫01ηλFϱ,λ+1ςωvp−upϱηϱ×ηup+1−ηvp1/p−1φ′ηup+1−ηvp1/pdη|≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×∫011−ηϱk+λ+ηϱk+ληup+1−ηvp1/p−1+η21−ηφ′u1−η2ηφ′vdη≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×∫011−ηϱk+ληup+1−ηvp1/p−1η21−ηφ′u1−η2ηφ′vdη+∫01ηϱk+ληup+1−ηvp1/p−1η21−ηφ′u1−η2ηφ′vdηφu+φv2−12Fϱ,λ+1ςωvp−uppvp−upλJϱ,λ,up+;ωςφoϕvpJϱ,λ,vp−;ωςφoϕup≤vp−up2pFϱ,λ+1ςωvp−upp|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×∫0112η1/21−ηϱk+λ−1/2φ′uηup+1−ηvp1/p−1dη+∫0112η−1/21−ηpk+λ+1/2φ′vηup+1−ηvp1/p−1dη+∫0112ηϱk+λ+1/21−η−1/2φ′uηup+1−ηvp1/p−1dη+∫0112ηϱk+λ−1/21−η1/2φ′vηup+1−ηvp1/p−1dη.
So,(16)φu+φv2−12Fϱ,λ+1ςωvp−uppvp−upλJϱ,λ,up+;ωςφoϕvpJϱ,λ,vp−;ωςφoϕup≤vp−up2pFϱ,λ+1ςωvp−upp|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×v1−p12β32,ϱk+λ+12F211−1p,32;ϱk+λ+2,1−upvpφ′u+v1−p12β12,ϱk+λ+322F11−1p,12;ϱk+λ+2,1−upvpφ′v+v1−p12βϱk+λ+32,122F11−1p,ϱk+λ+32;ϱk+λ+2,1−upvpφ′u+v1−p12βϱk+λ+12,322F11−1p,ϱk+λ+12;ϱk+λ+2,1−upvpφ′v.
From here,(17)φu+φv2−12Fϱ,λ+1ςωvp−uppvp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤vp−up2pFϱ,λ+1ςωvp−upp|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×v1−pϱk+λ−1/23ϱk+λ+12F11−1p,32;ϱk+λ+2,1−upvpφ′u+v1−pϱk+λ+1/2ϱk+λ+12F11−1p,12;ϱk+λ+2,1−upvpφ′v+v1−p5/4ϱk+λ+3/2ϱk+λ+12F11−1p,ϱk+λ+32;ϱk+λ+2,1−upvpφ′u+v1−p14ϱk+λ+1/2ϱk+λ+12F11−1p,ϱk+λ+12;ϱk+λ+2,1−upvpφ′v.
Simple calculations yield equation (13).Remark 2.
In Theorem1, we see the following:(1)
Forp=1, we have the inequality for MT-convex function:(18)φu+φv2−12Fϱ,λ+1ςωv−uϱv−uλvJϱ,λ,u+;ωςφoϕ+uJϱ,λ,v−;ωςφoϕ≤Fϱ,λ+1C1ωv−uϱ+Fϱ,λ+1C3ωv−uϱφ′u+Fϱ,λ+1C2ωv−uϱ+Fϱ,λ+1C4ωv−uϱφ′v.(2)
Forp=−1, we have the inequality of harmonically MT-convex function:(19)φu+φv2−12Fϱ,λ+1ςωu−v/uvϱu−v/uvλ1vJϱ,λ,1/u+;ω,ςφoϕ+1uJϱ,λ,1/vςφoϕ≤Fϱ,λ+1C1ωu−vuvϱ+Fϱ,λ+1C3ωu−vuvϱφ′u+Fϱ,λ+1C2ωu−vuvϱ+Fϱ,λ+1C4ωu−vuvϱφ′v.Theorem 2.
Letλ, ∈, ℝ+, φ: I⊂ℝ⟶ℝ, be a MT-non-convex function on u, v, ∈I such that u < v. If φ∈L[u, v], p > 0, then we get(20)φ12up+12vp1/p≤12Fϱ,λ+1ωvp−upϱvp−upλςJϱ,λ,up+;ωφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤14η1−ηφu+14η1−ηφ′v,where ϕc=c1/p.Proof.
Sinceφ is MT-non-convex on [u, v], for all c, d∈[u, v],(21)φ12cp+12dp1/p≤φc+φd2,and substituting c=ξup+1−ηvp1/p and d=1−ηup+ηvp1/p, equation (21) yields(22)φ12up+12vp1/p≤φηup+1−ηvp1/p+φ1−ηup+ηvp1/p.
Multiply inequality equation (22) by ηλ−1Fϱ,λςωvp−upϱηϱ, and after that, integrating it over η∈[0, 1], then we get(23)2Fϱ,λ+1ςωvp−upϱηϱφ12up+12vp1/p≤∫01ηλ−1Fϱ,λςωvp−upϱηϱφηup+1−ηvp1/pdη+∫01ηλ−1Fϱ,λςωvp−upϱηϱφ1−ηup+ηvp1/pdη=∫01ηλ−1∑k=0∞ςkωkvp−upϱkηϱkΓϱk+λφηup+1−ηvp1/pdη+∫01ηλ−1∑k=0∞ςkωkvp−upϱkηϱkΓϱk+λφ1−ηup+ηvp1/pdη=∫01∑k=0∞ςkωkΓϱk+ληϱk+λ−1φηup+1−ηvp1/pvp−upϱkdη+∫01∑k=0∞ςkωkΓϱk+ληϱk+λ−1φ1−ηup+ηvp1/pvp−upϱkdη=∫upvp∑k=0∞ςkωkΓϱk+λvp−cϱkvp−cλ−1vp−upϱkvp−upλ−1vp−upϱkφc1/pdcvp−up+∫upvp∑k=0∞ςkωkΓϱk+λc−upϱkc−upλ−1vp−upϱkvp−upλ−1vp−upϱkφc1/pdcvp−up=1vp−upλ∫upvp∑k=0∞ςkωkvp−cϱkΓϱk+λvp−cλ−1φc1/pdc+1vp−upλ∫upvp∑k=0∞ςkωkc−upϱkΓϱk+λc−upλ−1φc1/pdc=1vp−upλJϱ,λ,up+;ωςφoϕvp+1vp−upλJϱ,λ,vp−;ωςφoϕup2Fϱ,λ+1ςωvp−uppφ12up+12vp1/p≤1vp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup,which is left side of inequality equation (23). Now we have to prove right-hand side of inequality equation (22); applying definition of MT-non-convexity of φ,(24)φηup+1−ηvp1/p≤η21−ηφu+1−η2ηφv,and(25)φ1−ηup+ηvp1/p≤η21−ηφu+1−η2ηφv,and by adding equations (24) and (25), we get(26)φηup+1−ηvp1/p+φ1−ηup+ηvp1/p≤12η1−ηφu+12η1−ηφv.
Multiply inequality equation (26) by ηλ−1Fϱ,λςωvp−upϱηϱ, and after that, integrating it over η∈ [0, 1], then we get(27)1vp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤2Fϱ,λ+1σωvp−upρ14η1−ηφu+14η1−ηφv.
Combining equations (23) and (27) completes equation (20).Remark 3.
In Theorem2, we see the following:(1)
Forp=1, we have inequality for MT-convex function:(28)φ12u+12v,≤12Fϱ,λ+1ςωv−uϱv−uλvJϱ,λ,u+;ω,ςφoϕ+uJϱ,λ,v−;ω,ςφoϕ,≤14η1−ηφu+14η1−ηφv.(2)
Forp=−1, we have the inequality of harmonically MT-convex function:(29)φu+v2uv,≤12Fϱ,λ+1ςωu−v/uvϱu−v/uvλ1vJϱ,λ,1/v+;ω,ςφoϕ+1uJϱ,λ,1/v−;ω,ςφoϕ,≤14η1−ηφu+14η1−ηφv.


## 3. Fractional Integral Inequalities for (p-q) Convex Functions

In this section, we will develop fractional integral inequality for (p-q) convex function.Theorem 3.
Letλ, ∈, ℝ+, φ: I, ⊆ℝ+⟶ℝ+, be a differentiable mapping on Iou,v∈Io such that x < y. If φ′ is (p-q) convex on u,v, p > 0, then we obtain(30)φu+φv2−12Fϱ,λ+1ςωvp−upϱvp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup,≤Fϱ,λ+1C1ωvp−upϱ+Fϱ,λ+1C3ωvp−upϱφ′u,+Fϱ,λ+1C2ωvp−upϱ+Fϱ,λ+1C4ωvp−upϱφ′v,where(31)C1=ςkv1−pϱk+λ1/q+1ϱk+λ+1/q+12F11−1p,1q+1,ϱk+λ+1,1−upvpC2=ςkv1−pϱk+λ+1/qϱk+λ+1/q+12F11−1p,ϱk+λ+1q+1,ϱk+λ+1q+1,1−upvpC3=ςkv1−p1ϱk+λ+1/q+12F11−1p,ϱk+λ+1q+1,1,1−upvpC4=ςkv1−p1/qϱk+λ+1ϱk+λ+1/q+12F11−1p,1,1q−1,1−upvp.Proof.
By making use of Lemma1 and (p-q)-convexity of φ′, we obtain(32)φu+φv2−12vp−upλFϱ,λ+1ςωvp−upϱJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup=vp−up2pFϱ,λ+1ςωyp−upϱ∫011−ηλFϱ,λ+1ςωvp−upϱ1−ηϱ×ηup+1−ηvp1/p−1φ′ηup+1−ηvp1/pdη+∫01ηλFϱ,λ+1ςωvp−upϱηϱ×ηup+1−ηvp1/p−1φ′ηup+1−ηvp1/pdη≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×∫01|1−ηϱk+λ+ηϱk+ληup+1−ηvp1/p−1ηφ′uq+1−ηφ′vq1/qdη.
Moreover, we observe that(33)≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×∫011−ηϱk+ληup+1−ηvp1/p−1ηφ′uq+1−ηφ′vq1/qdη+∫01ηϱk+ληup+1−ηvp1/p−1ηφ′uq+1−ηφ′vq1/qdη,so that(34)φu+φv2−12vp−upλFϱ,λ+1ςωvp−upρJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×∫01η1/q1−ηϱk+λφ′uq1/qηup+1−ηvp1/p−1dη+∫011−ηϱk+λ+1/qφ′vq1/qηup+1−ηvp1/p−1dη+∫01ηϱk+λ+1/qφ′uq1/qηup+1−ηvp1/p−1dη+∫01ηϱk+λ1−η1/qφ′vq1/qηup+1−ηvp1/p−1dη.
Now(35)φu+φv2−12vp−upλFϱ,λ+1ςωvp−upρJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×v1−pβ1q+1,ϱk+λ+12F11−1p,1q+1,ϱk+λ+1q+2,1−upvpφ′u+v1−pβ1,ϱk+λ+1q+12F11−1p,1,ϱk+λ+1q+2,1−upvpφ′v+v1−pβϱk+λ+1q+1,12F11−1p,ϱk+λ+1q+1,ϱk+λ+1q+2,1−1pφ′u+v1−pβϱk+λ+1,1q+12F11−1p,ϱk+λ+1,ϱk+λ+1q+2,1−upvpφ′v.
From here,(36)φu+φv2−12vp−upλFϱ,λ+1ςωvp−upρJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1×v1−pϱk+λ1/q+1ϱk+λ+1/q+12F11−1p,1q+1,ϱk+λ+1q+2,1−upvpφ′u+v1−pϱk+λ+1/qϱk+λ+1/q+12F11−1p,1,ϱk+λ+1q+2,1−upvpφ′v+v1−p1ϱk+λ+1/q+12F11−1p,ϱk+λ+1q+1,ϱk+λ+1q+2,1−upvpφ′u+v1−p1/qϱk+λ+1ϱk+λ+1/q+12F11−1p,ϱk+λ+1,ϱk+λ+1q+2,1−upvpφ′v,and(37)φu+φv2−12vp−upλFϱ,λ+1ςωvp−upϱJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤vp−up2pFϱ,λ+1ςωvp−upϱ|∑k=0∞ςkωkvp−upϱkΓϱk+λ+1C1φ′u+C2φ′v+C3φ′u+C4φ′v,which is the required solution.Remark 4.
(1)
If one putsq=1 in equation (30), one has Theorem 5 in [22].(2)
Similarly, forq=1 and p=1 in equation (30), we get classical fractional integral of Hermite–Hadamard inequality.Theorem 4.
Letλ∈ℝ+, φ: I⊆ℝ⟶ℝ, be a (p-q) convex function on u, v, ∈I with u < v; if f∈L[u, v], p > 0, then we obtain(38)φ12up+12vp1/p≤12Fϱ,λ+1ωvp−upϱvp−upλςJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤φuq1/q+φvq1/q2,where gc=c1/p.Proof.
Sinceφ is (p-q) convex on [u, v], for all c, d∈[u, v],(39)φ12cp+12dp1/p≤12φcq+φdq1/q,and substituting c=ηup+1−ηvp1/p and d=1−ηup+ηvp1/p, then equation (39) yields(40)2φ12up+12vp1/p≤φηup+1−ηvp1/pq1/q+φ1−ηup+ηvp1/pq1/q.
Multiply inequality equation (40) by ηλ−1Fϱ,λςωvp−upϱηϱ, and after that, integrating over η∈[0, 1], we get(41)2Fϱ,λ+1ςωvp−upϱηϱφ12up+12vp1/p≤∫01ηλ−1Fϱ,λςωvp−upϱηϱφηup+1−ηvp1/pq1/qdη+∫01ηλ−1Fϱ,λςωvp−upϱηϱφ1−ηup+ηvp1/pq1/qdη=∫01ηλ−1∑k=0∞ςkωkvp−upϱkηϱkΓϱk+λφηup+1−ηvp1/pq1/qdη+∫01ηλ−1∑k=0∞ςkωkvp−upϱkηϱkΓϱk+λφ1−ηup+ηvp1/pq1/qdη=∫01∑k=0∞ςkωkΓϱk+ληϱk+λ−1φηup+1−ηvp1/pq1/qvp−upϱkdη+∫01∑k=0∞ςkωkΓϱk+ληϱk+λ−1φ1−ηup+ηvp1/pq1/qvp−upρkdη=∫upvp∑k=0∞ςkωkΓϱk+λvp−cϱkvp−cλ−1vp−upϱkvp−upλ−1vp−upϱkφc1/pq1/qdcvp−up+∫upvp∑k=0∞ςkωkΓϱk+λc−upϱkc−upλ−1vp−upϱkvp−upλ−1vp−upϱkφc1/pq1/qdcvp−up=1vp−upλ∫upvp∑k=0∞ςkωkvp−cϱkΓϱk+λvp−cλ−1φc1/pq1/qdc+1vp−upλ∫upvp∑k=0∞ςkωkc−upϱkΓϱk+λc−upλ−1φc1/pq1/qdc=1vp−upλJϱ,λ,up+;ωςφoϕvp+1vp−upλJϱ,λ,vp−;ωςφoϕup.
So, we have left-hand side of inequality equation (38).(42)2Fϱ,λ+1ςωvp−uppφ12up+12vp1/p≤1vp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup.
Now we have to prove other side of equation (40) from pq-convexity of φ.(43)φηup+1−ηvp1/p≤ηφuq+1−ηφvq1/q,and(44)φ1−ηup+ηvp1/p≤1−ηφuq+ηφvq1/q,and by adding inequality equations (43) and (44), we get(45)φηup+1−ηvp1/p+φ1−ηup+ηvp1/p≤ηφuq+1−ηφvq1/q+1−ηφuq+ηφvq1/qφηup+1−ηvp1/p+φ1−ηup+ηvp1/p≤φuq1/q+φvq1/q.
Multiply inequality equation (45) by ηλ−1Fϱ,λςωvp−upϱηϱ, and after that, integrating inequality over η∈[0, 1], we get(46)1vp−upλJϱ,λ,up+;ωςφoϕvp+Jϱ,λ,vp−;ωςφoϕup≤2Fϱ,λ+1ςωvp−upϱφuq1/q+φvq1/q2.
Combining equations (42) and (45) completes equation (38).Remark 5.
(1)
If one putsq=1 in equation (39), one has [22, Theorem 5].(2)
Similarly, forq=1 and p=1 in equation (39), we get classical fractional integral of H-H inequality.


## 4. Conclusion

Fractional integral inequalities are derived for MT-non-convex functions andp−q convex functions. With the help of several lemmas, the integral inequalities are derived in generalized fractional integral operator. The remarks at the end are also given to verify the extension of results.



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