# Positive Weighted Symmetry Function Kernels and Some Related Inequalities for a Generalized Class of Convex Functions

Authors: Yeliang Xiao; Ahsan Fareed Shah; Tariq Javed Zia; Ebenezer Bonyah
Journal: Journal of Function Spaces
 (2022)

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## Abstract

The fractional integral inequalities are crucial to deal applied problems. The present paper deals with the generalize midpoint type inequalities for a certain class of convex functions, namely, MT-convex functions in the setting of weighted fractional integral. It can be observed from the remarks given in the paper that our results or more generalized than existing results of literature and many of them can be obtained immediately from our results.


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## Body

## 1. Introduction

Through this paper, givenχ1,χ2 is a convex subset of Rˇ, a continuous function I:χ1,χ2⟶Rˇ is said to be convex on χ1,χ2, if the following inequality holds:
(1)Iŧχ1+1−ŧχ2≤ŧIχ1+1−ŧIχ2,where ŧ∈0,1.Fractional calculus is one of the interesting generalizations of classical calculus, as most of the differential equations are fractional in modeling real world problems [1–3]. So the fractional integral inequalities are the interesting area of research for many researchers of analysis and inequality [4, 5]. The operates involving fractional integrals become a hot area of research for the researcher working in inequality theory (see for example [5] and references therein). Many fractional integral operators are introduced in recent years, and every operator has its own certain properties, for example, Riemann-Liouville integral operator ([6, 7]), Caputo operator [8], Hadamard operator [9], Caputo Fabrizio operator [10], and Saigo operator [11].Like fractional calculus, fuzzy theory is also a brand new area of research. The concept of fuzzy set was developed in 1965. Since then, researchers have used this key set in many disciplines. Fuzzy set has also become an acknowledged research subject in both pure and applied mathematics and statistics, showing how this theory is highly applicable and productive in many applications [12–14]. Inequality related to fuzzy function is very papular in current years. One of the main and most powerful inequality is Hermite-Hadamard (H-H) type inequality [15], which is stated as follows:SupposeI:χ1,χ2⊆Rˇ⟶Rˇ is a convex function; then,
(2)Iχ1+χ22≤1χ2−χ1∫χ1χ2Indn≤Iχ1+Iχ22,with, χ1<χ2.For a further study of Hermite-Hadamard type inequality, read [16, 17].Definition 1 (see [5, 18]).
Letχ1,χ2⊂Rˇ and ℘χ be a monotone, positive and increasing function defined on χ1,χ2 having continuous derivative ℘′χ on χ1,χ2. Then left sided and right sided weighted fractional integrals for I, with respect to the function ℘χ on χ1,χ2, are stated as
(3)Iℏa;℘χ1+Iχ=ℏχ−1Γa∫χ1χ℘′ŧ℘χ−℘ŧa−1Iŧℏŧdŧ,Iχ2−a;℘ℏIχ=ℏχ−1Γa∫χχ2℘′ŧ℘ŧ−℘χa−1Iŧℏŧdŧ,a>0,where ℏχ−1≔1/ℏχ such that ℏχ≠0.The midpoint type inequality is also a H-H type inequality, and this inequality in the setting of RL and weighted fractional integrals with positive weighted symmetric kernelbu using the mid-pointχ1+χ2/2 of the given interval are presented in [19, 20], respectively.
(4)Iχ1+χ22≤2a−1Γa+1χ2−χ1aIχ1+χ2/2+aIχ2+Iχ1+χ2/2−aIχ1≤Iχ1+Iχ22,where I:χ1,χ2⟶Rˇ is convex and continuous
(5)Iχ1+χ22Ia;℘℘−1χ1+χ2/2+ℏo℘℘−1χ2+I℘−1χ1+χ2/2−a;℘ℏo℘℘−1χ1≤ℏχ2Iℏo℘a;℘℘−1χ1+χ2/2+Io℘℘−1χ2+ℏχ1I℘−1χ1+χ2/2−a;℘ℏo℘Io℘℘−1χ1≤Iχ1+Iχ22Ia;℘℘−1χ1+χ2/2+ℏo℘℘−1χ2+I℘−1χ1+χ2/2−a;℘ℏo℘℘−1χ1,where I:χ1,χ2⟶Rˇ is continuous and convex function and ℘χ is self increasing and monotone on χ1,χ2 with ℘′ is continuous function on χ1,χ2 and ℏ:χ1,χ2⟶0,∞ is an symmetric integrable function w.r.t χ1+χ2/2, where χ1<χ2 [5].This paper is organized as follows: initially, we will give some preliminaries material related to our research (Fractional Hermite-Hadamard-Fejer mid-point inequalities for a MT-Convex Function via an increasing function involving a positive weighted symmetric function). In the third section, we will present our main results.


## 2. Preliminaries

This section contains basic definitions and known results.Definition 2 (see [5, 21]).
Let a functionℏ;χ1,χ2⟶0,∞. We say that ℏ is symmetric w.r.t χ1+χ2/2 if we have
(6)ℏχ1+χ2−χ=ℏχ.Fejér [22] introduced the H-H-Fejer inequality by using the Definition 2, which is the weighted version of the inequality (2).
(7)Iχ1+χ22∫χ1χ2ℏχdχ≤∫χ1χ2Iχℏχdχ≤Iχ1+Iχ22∫χ1χ2ℏχdχ,where ℏ is the integrable function.Definition 3 (see [6, 7]).
Iχ1+aIχ and Iχ2−aIχ are the left- and right-sided RL fractional integrals of order a>0 defined by
(8)Iχ1+aIχ=1Γa∫χ1χχ−ŧa−1Iŧdŧ,χ>χ1,Iχ2−aIχ=1Γa∫χχ2ŧ−χa−1Iŧdŧ,χ<χ2.The end-point inequalities of H-H type in the setting of RL fractional integrals are presented in [23, 24].
(9)Iχ1+χ22≤Γa+12χ2−χ1aIχ1+aIχ2+Iχ2−aIχ1≤Iχ1+Iχ22Iχ1+χ22Iχ1+aℏχ2+Iχ2−aℏχ1≤Iχ1+aIℏχ2+Iχ2−aIℏχ1≤Iχ1+Iχ22Iχ1+aℏχ2+Iχ2−aℏχ1,where I is positive convex and continuous function on χ1,χ2, when Iχ∈L1χ1,χ2 and χ1<χ2.Definition 4 (see [25]).
A functionI:χ1,χ2⊆Rˇ⟶χ1,χ2 is declared to be MT-convex on χ1,χ2, if the following inequality holds:
(10)Iŧχ1+1−ŧχ2≤ŧ21−ŧIχ1+1−ŧ2ŧIχ2,where ŧ∈0,1.Theorem 5.
Letχ2>χ1≥0; let I:χ1,χ2⟶R be an L1 MT-convex function and the weighted symmetric positive function ℏ:χ1,χ2⟶R w.r.t χ1+χ2/2. In addition, the function ℘ is as in Definition 1; then for a>0, we have
(11)Iχ1+χ22Ia:℘℘−1χ1+χ2/2+ℏ∘℘℘−1χ2+I℘−1χ1+χ2/2−a:℘ℏ∘℘℘−1χ1≤ℏχ2Iℏ∘℘a:℘℘−1χ1+χ2/2+I∘℘℘−1χ2+ℏχ1I℘−1χ1+χ2/2−a:℘ℏ∘℘I∘℘℘−1χ1≤Iχ1+Iχ2χ2−χ1222ΓaA11awhere
(12)A11a=∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa−3/2℘ŧ−χ1−1/2ℏo℘ŧ℘′ŧdŧProof.
The reader can prove first inequality as proved in ([20] Theorem 1).
For the second inequality, using MT-convexity ofI, we can get
(13)Iŧ2χ1+2−ŧ2χ2+I2−ŧ2χ1+ŧ2χ2≤Iχ1+Iχ21ŧ2−ŧ.
Multiply both sides byŧa−1ℏŧ/2χ1+2−ŧ/2χ2(14)ŧa−1ℏŧ2χ1+2−ŧ2χ2Iŧ2χ1+2−ŧ2χ2+ŧa−1ℏŧ2χ1+2−ŧ2χ2I2−ŧ2χ1+ŧ2χ2≤Iχ1+Iχ2ŧa−3/22−ŧℏŧ2l+2−ŧ2χ2.
Integrating both sides with respect toŧ over [0,1], we get
(15)∫01ŧa−1ℏŧ2χ1+2−ŧ2χ2Iŧ2χ1+2−ŧ2χ2dŧ+∫01ŧa−1ℏŧ2χ1+2−ŧ2χ2I2−ŧ2χ1+ŧ2χ2dŧ≤Iχ1+Iχ2∫01ŧa−3/22−ŧℏŧ2χ1+2−ŧ2χ2dŧ.
From ([20] Theorem 1), we can write left hand side of the above as
(16)∫01ŧa−1ℏŧ2χ1+2−ŧ2χ2Iŧ2χ1+2−ŧ2χ2dŧ+∫01ŧa−1ℏŧ2χ1+2−ŧ2χ2I2−ŧ2χ1+ŧ2χ2dŧ=2aΓaχ2−χ1aℏχ2Iℏ∘℘a:℘℘−1χ1+χ2/2+I∘℘℘−1χ2+ℏχ1I℘−1χ1+χ2/2−a:℘ℏ∘℘I∘℘℘−1χ1,taking ŧ=2χ2−℘ŧ/χ2−χ1(17)ℏχ2℘−1χ1+χ2/2+Iℏ∘℘a:℘I∘℘℘−1χ2+ℏχ1ℏ∘℘I℘−1χ1+χ2/2−a:℘I∘℘℘−1χ1≤Iχ1+Iχ2χ2−χ1a2aΓa∫01ŧa−3/22−ŧℏŧ2χ1+2−ŧ2χ2dŧ≤Iχ1+Iχ2χ2−χ1222ΓaA11a,where
(18)A11a=∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa−3/2℘ŧ−χ1−1/2ℏo℘ŧ℘′ŧdŧ,which completes the proof. From Theorem 5, we can get following remarks as follows.Remark 6.
When℘χ=χ, then 12 becomes
(19)Iχ1+χ22Iχ1+χ2/2+RLahχ2+RLIχ1+χ2/2−aℏχ1≤ℏχ2Iχ1+χ2/2+RLℏaIχ2+ℏχ1IℏRLχ1+χ2/2−aIχ1≤Iχ1+Iχ22χ2−χ122Γa∫χ1+χ2/2χ2χ2−ŧa−3/2ŧ−χ1−1/2ℏŧdŧ,where χ1+RLIℏa and ℏRLIχ2−a are left- and right-weighted RL fractional integrals, respectively,
(20)Iχ1+RLℏaIχ=ℏ−1χΓa∫χ1χχ−za−1Izℏzdz,IℏRLχ2−aIχ=h−1χΓa∫χχ2z−χa−1Izℏzdz,a>0.Remark 7.
When℘χ=χ and a=1, then 12 becomes
(21)Iχ1+χ22∫χ1χ2ℏχdχ≤∫χ1χ2Iχℏχdχ≤Iχ1+Iχ22χ2−χ122∫χ1+χ2/2χ2χ2−ŧŧ−χ1−1/2ℏŧdŧ.Remark 8.
When℘χ=χ and ℏχ=1, then 12 becomes
(22)Iχ1+χ22≤2a−1Γa+1χ2−χ1aIχ1+χ2/2+aIχ2+Iχ1+χ2/2−aIχ1≤Iχ1+Iχ22χ2−χ122Γa∫χ1+χ2/2χ2χ2−ŧa−3/2ŧ−χ1−1/2dŧ.Remark 9.
When℘x=x, a=1 and ℏx=1, then 12 becomes
(23)Iχ1+χ22≤1χ2−χ1∫χ1χ2Indn≤Iχ1+Iχ28πχ2−χ12.Lemma 10 (see [20] Lemma 2).
Letχ2≥χ1>0, let I:χ1,χ2⟶R be a continuous function with a derivative I′∈L1χ1,χ2 such that
(24)Iχ=Iχ1+∫χ1χI′tdt‍,and let ℏ:χ1,χ2⟶R is as in Theorem 5 and ℘ is as in Definition 1; then, for a>0, we have
(25)Iχ1+χ22I℘−1χ1+χ22+a:℘ℏ∘℘℘−1χ2+I℘−1χ1+χ2/2−a:℘ℏ∘℘℘−1χ1−ℏχ2I℘−1χ1+χ22+ℏ∘℘a:℘I∘℘℘−1χ2+ℏχ1ℏ∘℘I℘−1χ1+m/2−a:℘I∘℘℘−1χ1=1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1ŧ℘′χ℘χ−χ1a−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ+−1Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ.
Letting(26)L1+L2=1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1ŧ℘′χ℘χ−χ1a−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ+−1Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2‍℘′χχ2−℘χa−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧimplies
(27)L1=Iχ1+χ22I℘−1χ1+χ2/2+a:℘ℏ∘℘℘−1χ2+I℘−1χ1+χ2/2−a:℘ℏ∘℘℘−1χ1L2=−ℏχ2℘−1χ1+χ2/2+Iℏ∘℘a:℘I∘℘℘−1χ2+ℏχ1ℏ∘℘I℘−1χ1+χ2/2−a:℘I∘℘℘−1χ1


## 3. Main Results

With the help of Lemma10, we establish H-H-F type inequalities for MT-convex functions in this section.Theorem 11.
Let0≤χ1<χ2; let I:χ1,χ2⊂0,∞⟶Rˇ be a (continuously) differentiable mapping on χ1,χ2 such that
(28)Iχ=Iχ1+∫χ1χI′ŧdŧ‍,and let ℏ:χ1,χ2⟶Rˇ be an integrable, positive, and weighted symmetric function with respect to χ1+χ2/2. If, in addition, I′ is MT-convex on χ1,χ2 and ℘ is an increasing and positive function from χ1,χ2 onto itself such that its derivative ℘′χ is continuous on χ1,χ2, then for a>0, the following inequalities hold:
(29)L1+L2=1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1ŧ℘′χ℘χ−χ1a−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ−1Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ≤AaI′χ1+BaI′χ2,where
(30)Aa=ℏχ1,χ1+χ2/2,∞2aΓa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a−1/2χ2−℘ŧ1/2℘′ŧdŧ+ℏχ1+χ2/2,χ2,∞2aΓa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa+1/2℘ŧ−χ1−1/2℘′ŧdŧ(31)Ba=ℏχ1,χ1+χ2/2,∞2aΓa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a+1/2χ2−℘ŧ−1/2℘′ŧdŧ+ℏχ1+χ2/2,χ2,∞2aΓa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa−1/2℘ŧ−χ11/2℘′ŧdŧ.Proof.
Using (26) and properties of modulus, we get
(32)L1+L2=1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ−1Γa∫℘−1χ1+χ2/2℘−1χ2∫t−℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ≤1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ+1Γa∫℘−1χ1+χ2/2℘−1χ2∫t−℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ.
Applying (10) on I′ for ŧ∈℘−1χ1,℘−1χ2, we get
(33)I′∘℘ŧ=I′χ2−℘ŧχ2−χ1χ1+℘ŧ−χ1χ2−χ1χ2≤12χ2−℘ŧ℘ŧ−χ1I′χ1+℘ŧ−χ1χ2−℘ŧI′χ2.
Hence, we obtain(34)L1+L2ℏχ1,χ1+χ2/2,∞2Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1dχ×χ2−℘℘χ−χ1℘℘χ−χ1−χ1I′χ1+℘℘χ−χ1−χ1χ2−℘℘χ−χ1I′χ2℘′℘χ−χ1d℘χ−χ1+ℏχ1+χ2/2,χ2,∞2Γa∫℘−1χ1+χ2/2℘−1χ2∫t−℘−1χ2℘′χχ2−℘χa−1dχ×χ2−℘ŧ℘ŧ−χ1I′χ1+℘ŧ−χ1χ2−℘ŧI′χ2℘′ŧdŧ≤ℏχ1,χ1+χ2/2,∞2aΓa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a×χ2−℘ŧ℘ŧ−χ1I′χ1+℘ŧ−χ1χ2−℘ŧI′χ2℘′ŧdŧ+ℏχ1+χ2/2,χ2,∞2aΓa∫℘−1χ1+χ2/2℘−1χ2χ2−℘χa×χ2−℘ŧ℘ŧ−χ1I′χ1+℘ŧ−χ1χ2−℘ŧI′χ2℘′ŧdŧ≤ℏχ1,χ1+χ2/2,∞2aΓaI′χ1∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a−1/2χ2−℘ŧ1/2℘′ŧdŧ+I′χ2∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a+1/2χ2−℘ŧ−1/2℘′ŧdŧ+ℏχ1+χ2/2,χ2,∞2aΓaI′χ1∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa+1/2℘ŧ−χ1−1/2℘′ŧdŧ+I′χ2∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa−1/2℘ŧ−χ11/2℘′ŧdŧ≤ℏχ1,χ1+χ2/2,∞2aΓa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a−1/2χ2−℘ŧ1/2℘′ŧdŧ+ℏχ1+χ2/2,χ2,∞2aΓa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa+1/2℘ŧ−χ1−1/2℘′ŧdŧI′χ1+ℏχ1,χ1+χ2/2,∞2aΓa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a+1/2χ2−℘ŧ−1/2℘′ŧdŧ+ℏχ1+χ2/2,χ2,∞2aΓa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa−1/2℘ŧ−χ11/2℘′ŧdŧI′χ2≤AaI′χ1+BaI′χ2.
From (30) and (31), we get required inequality. From Theorem 11, we can get following remarks as follows.Remark 12.
When℘χ=χ, then inequality (29) becomes
(35)Iχ1+χ22Iχ1+χ2/2+RLaℏχ2+IRLχ1+χ2/2−aℏχ1−ℏχ2Iχ1+χ2/2+RLℏaIχ2+ℏχ1IℏRLχ1+χ2/2−aIχ1⩽A1I′χ1+A2I′χ2,where
(36)A1a=ℏχ1,χ1+χ2/2,∞2aΓa∫χ1χ1+χ2/2ŧ−χ1a−1/2χ2−ŧ1/2dŧ+ℏχ1+χ2/2,χ2,∞2aΓa∫χ1+χ2/2χ2χ2−ŧa+1/2ŧ−χ1−1/2dŧA2a=ℏχ1,χ1+χ2/2,∞2aΓa∫χ1χ1+χ2/2ŧ−χ1a+1/2χ2−ŧ−1/2dŧ+ℏχ1+χ2/2,χ2,∞2aΓa∫χ1+χ2/2χ2χ2−ŧa−1/2ŧ−χ11/2dŧ.Remark 13.
When℘χ=χ and ℏχ=1, then inequality (29) becomes
(37)2a−1Γa+1χ2−χ1aIχ1+χ2/2+RLaIχ2+IRLχ1+χ2/2−aIχ1−Iχ1+χ22≤A3I′χ1+A4I′χ2,where
(38)A3a=12aΓa∫χ1χ1+χ2/2ŧ−χ1a−1/2χ2−ŧ1/2dŧ+12aΓa∫χ1+χ2/2χ2χ2−ŧa+1/2ŧ−χ1−1/2dŧA4a=12aΓa∫χ1χ1+χ2/2ŧ−χ1a+1/2χ2−ŧ−1/2dŧ+12aΓa∫χ1+χ2/2χ2χ2−ŧa−1/2ŧ−χ11/2dŧ.Remark 14.
When℘χ=χ, a=1, and ℏχ=1, then inequality (29) becomes
(39)1χ2−χ1∫χ1χ2Iχdχ−Iχ1+χ22χ2−χ128πI′χ1+I′χ2.Theorem 15.
Let0χ1<χ2; let I:χ1,χ2⊂0,∞⟶R˘ be a (continuously) differentiable mapping on χ1,χ2 such that
(40)Iχ=Iχ1+∫χ1χI′ŧdŧ,and let ℏ:χ1,χ2⟶R˘ as in Theorem 5. IfI′r is MT-convex stratifying the conditions of Definition 1, then for a>0, we have
(41)L1+L2≤χ2−χ1a+1,1−1/r2a1−1/r+1aa+11−1/rΓa×CaI′χ1r+DaI′χ2r+EaI′χ1r+FaI′χ2r,where
(42)Ca=ℏχ1,χ1+χ2/2,∞∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a−1/2χ2−℘ŧ1/2℘′ŧdŧ,(43)Da=ℏχ1,χ1+χ2/2,∞∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a+1/2χ2−℘ŧ−1/2℘′ŧdŧ,(44)Ea=ℏχ1+χ2/2,χ2,∞∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa+1/2℘ŧ−χ1−1/2℘′ŧdŧ,(45)Fa=ℏχ1+χ2/2,χ2,∞∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa−1/2℘ŧ−χ11/2℘′ŧdŧ.Proof.
By (10) on I′r for ŧ∈℘−1χ1,℘−1χ2, we get
(46)I′∘℘ŧr=I′χ2−℘ŧχ2−χ1χ1+℘ŧ−χ1χ2−χ1χ2r≤12χ2−℘ŧ℘ŧ−χ1I′χ1r+℘ŧ−χ1χ2−℘ŧI′χ2r.
Using (26), power mean inequality and MT-convexity of I′r(47)L1+L2≤1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1ℏ∘℘χdχI′∘℘ŧ℘′ŧdŧ+1Γa∫℘−1χ1+m/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχI′∘℘ŧ℘′ŧdŧ≤1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1ŧ℘′χ℘χ−χ1a−1ℏ∘℘χdχ℘′ŧdŧ1−1/r×∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1ℏ∘℘χdχI′∘℘ŧr℘′ŧdŧ1/r+1Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχ℘′ŧdŧ1−1/r×∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχI′∘℘ŧr℘′ŧdŧ1/r≤ℏχ1,χ1+χ2/2,∞Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1dχ℘′ŧdŧ1−1/r×∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1dχI′∘℘ŧr℘′ŧdŧ1/r+ℏχ1+χ2/2,χ2,∞Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1dχ℘′ŧdŧ1−1/r×∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1dχI′∘℘ŧr℘′ŧdŧ1/r≤ℏχ1,χ1+χ2/2,∞21/rΓa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1aa℘′ŧdŧ1−1/r×∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1aaχ2−℘ŧ℘ŧ−χ1I′χ1r+℘ŧ−χ1χ2−℘ŧI′χ2r℘′ŧdŧ1/r+ℏχ1+χ2/2,χ2,∞21/rΓa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧaa℘′ŧdŧ1−1/r×∫℘−1χ1+χ2/2℘−1χ2χ2−℘taaχ2−℘ŧ℘ŧ−χ1I′χ1r+℘ŧ−χ1χ2−℘ŧI′χ2r℘′ŧdŧ1/r≤ℏχ1,χ1+χ2/2,∞21/2aa+11−1/rΓaχ2−χ12a+11−1/r×I′χ1r∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a−1/2χ2−℘ŧ1/2℘′ŧdŧ+I′χ2r∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1a+1/2χ2−℘ŧ−1/2℘′ŧdŧ1/r+ℏχ1+χ2/2,χ2,∞21/raa+11−1/rΓaχ2−χ12a+11−1/r×I′χ1r∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa+1/2℘ŧ−χ1−1/2℘′ŧdŧ+I′χ2r∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧa−1/2℘ŧ−χ11/2℘′ŧdŧ1/r,from (42), (43), (44), and (45), we get the required inequality.
(48)⩽χ2−χ1a+11−1/r2a1−1/r+1aa+11−1/rΓaCaI′χ1r+DaI′χ2r1/r+EaI′χ1r+FaI′χ2r1/r.From Theorem15, we can get following remarks as follows:Remark 16.
When℘χ=χ, then inequality (41) becomes
(49)Iχ1+χ22Iχ1+χ2/2+RLaℏχ2+RLIχ1+χ2/2−aℏχ1−ℏχ2Iχ1+χ2/2+RLℏaIχ2+ℏχ1IℏRLχ1+χ2/2−aIχ1,≤χ2−χ1a+11−1/r2a1−1/r+1aa+11−1/rΓa×C1aI′χ1r+D1aI′χ2r1/r+E1aI′χ1r+F1aI′χ2r1/r,where
(50)C1a=ℏχ1,χ1+χ2/2,∞∫χ1χ1+χ2/2ŧ−χ1a−1/2χ2−ŧ1/2dŧ,D1a=ℏχ1,χ1+χ2/2,∞∫χ1χ1+χ2/2ŧ−χ1a+1/2χ2−ŧ−1/2dŧ,E1a=ℏχ1+χ2/2,χ2,∞∫χ1+χ2/2χ2χ2−ŧa+1/2ŧ−χ1−1/2dŧ,F1a=ℏχ1+χ2/2,χ2,∞∫χ1+χ2/2χ2χ2−ŧa−1/2ŧ−χ11/2dŧ.Remark 17.
When℘χ=χ and ℏχ=1, then inequality (41) becomes
(51)2a−1Γa+1χ2−χ1aIχ1+χ2/2+RLaIχ2+IRLχ1+χ2/2−aIχ1−Iχ1+χ22≤χ2−χ1a+11−1/r2a1−1/r+1aa+11−1/rΓaC2aI′χ1r+D2aI′χ2r1/r+E2aI′χ1r+F2aI′χ2r1/r,where
(52)C2a=∫χ1χ1+χ2/2ŧ−χ1a−1/2χ2−ŧ1/2dŧ,D2a=∫χ1χ1+χ2/2ŧ−χ1a+1/2χ2−ŧ−1/2dŧ,E2a=∫χ1+χ2/2χ2χ2−ŧa+1/2ŧ−χ1−1/2dŧ,F2a=∫χ1+χ2/2χ2χ2−ŧa−1/2ŧ−χ11/2dŧ.Remark 18.
When℘χ=χ, a=1 and ℏχ=1, then inequality (41) becomes
(53)1χ2−χ1∫χ1χ2Iχdχ−Iχ1+χ22≤χ2−χ12−2/r23−2/r×πχ2−χ1216I′χ1r+3π−8χ2−χ1216I′χ2r1/r+3π−8χ2−χ1216I′χ1r+πχ2−χ1216I′χ2r1/r.Theorem 19.
Let0χ1<χ2; let I:χ1,χ2⊂0,∞⟶R˘ be a (continuously) differentiable mapping on χ1,χ2 such that
(54)Iχ=Iχ1+∫χ1χI′ŧdŧ,and let ℏ:χ1,χ2⟶R˘ be an integrable, positive, and weighted symmetric function with respect to χ1+χ2/2. If, in addition, I′r is MT-convex on χ1,χ2 with 1/v+1/r=1 and r>1 and ℘ is an increasing and positive function from χ1,χ2 onto itself such that its derivative ℘′χ is continuous on χ1,χ2, then for a>0, the following inequalities hold:
(55)L1+L2χ2−χ1a+1/u21/r+a+1/uauau+11/u×KaI′χ1r+LaI′χ2r1/r+MaI′χ1r+NaI′χ2r1/r,where
(56)Ka=ℏχ1,χ1+χ2/2,∞Γa∫℘−1χ1℘−1χ1+χ2/2χ2−℘ŧ℘ŧ−χ1℘′ŧdŧ,(57)La=ℏχ1,χ1+χ2/2,∞Γa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1χ2−℘ŧ℘′ŧdŧ,(58)Ma=ℏχ1+χ2/2,χ2,∞Γa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧ℘ŧ−χ1℘′ŧdŧ,(59)Na=ℏχ1+χ2/2,χ2,∞Γa∫℘−1χ1+χ2/2℘−1χ2℘ŧ−χ1χ2−℘ŧ℘′ŧdŧ.Proof.
By (10) onI′r for ŧ∈℘−1χ1,℘−1χ2, we get
(60)I′∘℘ŧr=I′χ2−℘ŧχ2−χ1χ1+℘ŧ−χ1χ2−χ1χ2r≤12χ2−℘ŧ℘ŧ−χ1I′χ1r+℘ŧ−χ1χ2−℘ŧI′χ2r.
Using (26), Holder inequality, MT-convexity of I′r, and properties of modulus, we get
(61)L1+L2≤1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ+1Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχ×I′∘℘ŧ℘′ŧdŧ≤1Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1t−℘′χ℘χ−χ1a−1ℏ∘℘χdχu℘′ŧdŧ1/u×∫℘−1χ1℘−1χ1+χ2/2I′∘℘ŧr℘′ŧdŧ1/r+1Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1ℏ∘℘χdχu℘′ŧdŧ1/u×∫℘−1χ1+χ2/2℘−1χ2I′∘℘ŧr℘′ŧdŧ1/r≤ℏχ1,χ1+χ2/2,∞Γa∫℘−1χ1℘−1χ1+χ2/2∫℘−1χ1ŧ℘′χ℘χ−χ1a−1dχu℘′ŧdŧ1/u×∫℘−1χ1℘−1χ1+χ2/2I′∘℘ŧr℘′ŧdŧ1/u+ℏχ1+χ2/2,χ2,∞Γa∫℘−1χ1+χ2/2℘−1χ2∫ŧ℘−1χ2℘′χχ2−℘χa−1dχu℘′ŧdŧ1/u×∫℘−1χ1+χ2/2℘−1χ2I′∘℘ŧr℘′ŧdŧ1/r≤ℏχ1,χ1+χ2/2,∞21/rauΓa∫℘−1χ1℘−1χ1+χ2/2℘ŧ−χ1au℘′ŧdŧ1/u×∫℘−1χ1℘−1χ1+χ2/2χ2−℘ŧ℘ŧ−χ1I′χ1r+℘ŧ−χ1χ2−℘ŧI′χ2r℘′ŧdŧ1/r+ℏχ1+χ2/2,χ2,∞21/rauΓa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧau℘′ŧdŧ1/u,×∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧ℘ŧ−χ1I′χ1r+℘ŧ−χ1χ2−℘ŧI′χ2r℘′ŧdŧ1/r≤χ2−χ1a+1/u21/r+a+1/uauau+11/uℏχ1,χ1+χ2/2,∞Γa×∫℘−1χ1℘−1χ1+χ2/2χ2−℘ŧ℘ŧ−χ1℘′ŧdŧI′χ1r+ℏχ1,χ1+χ2/2,∞Γa∫℘−1χ1℘−1℘ŧ−χ1χ2−℘ŧ℘′ŧdŧI′χ2r1/r+ℏχ1+χ2/2,χ2,∞Γa∫℘−1χ1+χ2/2℘−1χ2χ2−℘ŧ℘ŧ−χ1℘′ŧdŧI′χ1r+ℏχ1+χ2/2,χ2,∞Γa∫℘−1χ1+χ2/2℘−1χ2℘ŧ−χ1χ2−℘ŧ℘′ŧdŧI′χ2r1/r,from (56); we get required inequality.
(62)χ2−χ1a+1/u21/r+a+1/uauau+11/u×KaI′χ1r+LaI′χ2r1/r+MaI′χ1r+NaI′χ2r1/r.From Theorem19, we can get following remarks as follows:Remark 20.
When℘χ=χ, then inequality (55) becomes Iχ1+χ2/2Iχ1+χ2/2+RLaℏχ2+RLIχ1+χ2/2−aℏχ1−ℏχ2Iχ1+χ2/2+RLℏaIχ2+ℏχ1IℏRLχ1+χ2/2−aχ2χ1⩽χ2−χ1a+1/u/21/r+a+1/uauau+11/u×K1aI′χ1r+L1aI′χ2r1/r+M1aI′χ1r+N1aI′χ2r1/r, (37) where
(63)K1a=ℏχ1,χ1+χ2/2,∞Γa∫χ1χ1+χ2/2χ2−ŧŧ−χ1dŧ,L1a=ℏχ1,χ1+χ2/2,∞Γa∫χ1χ1+χ2/2ŧ−χ1χ2−ŧdŧ,M1a=ℏχ1+χ2/2,χ2,∞Γa∫χ1+χ2/2χ2χ2−ŧŧ−χ1dŧ,N1a=ℏχ1+χ2/2,χ2,∞Γa∫χ1+χ2/2χ2ŧ−χ1χ2−ŧdŧ.Remark 21.
When℘χ=χ and ℏχ=1, then inequality (55) becomes
(64)2a−1Γa+1χ2−χ1aIχ1+χ2/2+RLaIχ2+IRLχ1+χ2/2−aIχ1−Iχ1+χ22≤χ2−χ1a+1/u21/r+a+1/uauau+11/u×K2aI′χ1r+L2aI′χ2r1/r+M2aI′χ1r+N2aI′χ2r1/r,where
(65)N2a=K2a=1Γaπ+2χ2−χ14,M2a=L2a=1Γaπ−2χ2−χ14.Remark 22.
When℘χ=χ, a=1 and ℏχ=1, then inequality (55) becomes
(66)1χ2−χ1∫χ1χ2Iχdχ−Iχ1+χ22χ2−χ11+1/u21/r+1+1/uu+11/u×K3I′χ1r+L3I′χ2r1/r+M3I′χ1r+N3I′χ2r1/r,where
(67)N3=K3=π+2χ2−χ14,M3=L3=π−2χ2−χ14.


## 4. Conclusions

In this paper, the generalized midpoint type inequalities for MT-convex functions in the setting of weighted fractional integral have been established. The results established in this paper extend and generalize many existing results which are explained in the remarks. Our results can be extended for more generalized class of convex and concave functions.



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