# Some New Kinds of Fractional Integral Inequalities via Refinedα,h−m-Convex Function

Authors: Moquddsa Zahra; Muhammad Ashraf; Ghulam Farid; Kamsing Nonlaopon
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

In this article, we present new integral inequalities for refinedα,h−m-convex functions using unified integral operators (12) and (13). The established results provide the refinements of several well-known integral and fractional integral inequalities.


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## Body

## 1. Introduction

Convex functions are important in diverse fields of mathematics, statistics, engineering, and optimization. Especially in the formation of inequalities, they play a very vital role. In the subject of mathematical analysis, inequalities provide a significant contribution in developing classical concepts and notions. For example, inequalities well known as Cauchy–Schwarz inequality, Chebyshev inequality, Minkowski inequality, Hadamard inequality, and Jensen inequality are utilized frequently in pure and applied mathematics. It is always a challenge to extend, generalize, and refine such inequalities by considering new classes of functions. In this era, researchers are working on classical inequalities concerning fractional integral and derivative operators. It can be observed that the Hadamard inequality is studied more for many kinds of fractional integral and derivative operators than any other classical inequality, see [1–7] for more details.The aim of this paper is to study the refinements of Hadamard and other integral inequalities recently studied in [8–11]. The consequences of these inequalities also provide refinements of fractional integral inequalities connected with the integral inequalities studied in the recent past.The article is organized as follows. In Section2, we suggest some preliminaries. In Section 3, the bounds of unified integral operators are given using refined α,h−m-convex functions. These are the refinements of bounds already obtained in the literature. In Section 4, some applications of the main results are given in the form of fractional integral inequalities and their refinements.


## 2. Preliminaries

In this section, we give definitions of different kinds of convex functions and integral operators which will be useful in formulating the results of this paper. Throughout the paper, all the functions are assumed to be real-valued functions until specified.Definition 1.
(see [12]). A function Ω is called convex if(1)Ωtx1′+1−ty1′≤tΩx1′+1−tΩy1′,holds for all x1′,y1′∈I⊆ℝ and t∈0,1.Definition 2.
(see [1]). A function Ω is called s,m-convex if for each x1′,y1′∈0,v⊆ℝ, we have(2)Ωtx1′+m1−ty1′≤tsΩx1′+m1−tsΩy1′,where t∈0,1 and s,m∈0,12.Definition 3.
(see [13]). A function Ω is called α,m-convex if for each x1′,y1′∈0,v⊆ℝ, we have(3)Ωtx1′+m1−ty1′≤tαΩx1′+m1−tαΩy1′,where α,m∈0,12 and t∈0,1.Definition 4.
(see [4]). Let h:J⟶ℝ is a function with h≡0 and 0,1⊆J. A function Ω is said to be h−m-convex, if Ω,h≥0 and for each x1′,y1′∈0,v⊆ℝ, we have(4)Ωtx1′+m1−ty1′≤htΩx1′+mh1−tΩy1′,where m∈0,1 and t∈0,1.Definition 5 (see [4]).
Leth:J⟶ℝ is a function with h≡0 and 0,1⊆J. A function Ω is said to be α,h−m-convex, if Ω,h≥0 and for each x1′,y1′∈0,v⊆ℝ, we have(5)Ωtx1′+m1−ty1′≤htαΩx1′+mh1−tαΩy1′,where α,m∈0,12 and t∈0,1.Definition 6.
(see [14]). Let h:J⟶ℝ be a function with h≡0 and 0,1⊆J. A function Ω is called refined α,h−m-convex function, if Ω,h≥0 and for each x1′,y1′∈0,v⊆ℝ, we have(6)Ωtx1′+m1−ty1′≤htαh1−tαΩx1′+mΩy1′,where α,m∈0,12 and t∈0,1.
Inequality (6) gives refinements of several types of convexities when 0<ht<1, see [14].
The need for integral operators in the study of fractional derivatives is of immense importance. In the recent era, integral operators are being used extensively for producing new results in the literature. For references, see [2, 4–6]. Next, we give some fundamental integral operators which are used in this paper.Definition 7.
(see [15]). Let Ω∈L1x1′,y1′ and Δ be positive and increasing function having a continuous derivative on x1′,y1′. The left and right fractional integrals of Ω with respect to Δ on x1′,y1′ of order κ are given by(7)ΔκIy1′+Ωx=1Γκ∫x1′xΔx−Δtκ−1Δ′tΩtdt,x>x1′,ΔκIy1′−Ωx=1Γκ∫xy1′Δt−Δxκ−1Δ′tΩtdt,x<y1′,where Γ. is the gamma function and ℜκ>0.Definition 8.
(see [16]). Let Ω∈L1x1′,y1′ and Δ be positive and increasing function having a continuous derivative on x1′,y1′. The left and right k-fractional integrals of Ω with respect to Δ on x1′,y1′ of order κ are given by(8)ΔκIx1′−Ωx=1kΓkκ∫x1′xΔx−Δtκ/k−1Δ′tΩtdt,x>x1′,(9)ΔκIy1′−Ωx=1kΓkκ∫xy1′Δt−Δxκ/k−1Δ′tΩtdt,x<y1′,where Γk. is the k-gamma function and ℜκ,k>0.Definition 9 (see [17]).
LetΩ∈L1x1′,y1′ and x∈x1′,y1′, also let σ,κ,α,ξ,γ,ι∈ℂ,ℜκ,ℜα,ℜξ>0,ℜι>ℜγ>0 with p≥0,δ>0, and 0<k≤δ+ℜκ, then the generalized fractional integral operators ϵκ,α,ξ,σ,x1′+γ,δ,k,ιΩ and ϵκ,α,ξ,σ,y1′−γ,δ,k,ιΩ are defined by(10)ϵκ,α,ξ,σ,x1′+γ,δ,k,ιΩx;p=∫x1′xx−tα−1Eκ,α,ξγ,δ,k,ισx−tκ;pΩtdt,ϵκ,α,ξ,σ,y1′−γ,δ,k,ιΩx;p=∫xy1′t−xα−1Eκ,α,ξγ,δ,k,ισt−xκ;pΩtdt,where Eκ,α,ξγ,δ,k,ιt;p is the extended generalized Mittag–Leffler function defined as(11)Eκ,α,ξγ,δ,k,ιt;p=∑n=0∞ρpγ+nk,ι−γργ,ι−γιnkΓκn+αtnξnδ.Definition 10 (see [16]).
LetΩ,Δ be real-valued functions defined over x1′,y1′ with 0<x1′<y1′, where Ω is positive and integrable and Δ is differentiable and strictly increasing. Also, let ϒ/x be an increasing function on x1′,∞ and α,ξ,γ,ι∈ℂ,p,κ,δ≥0, and 0<k≤δ+κ. Then, for x∈x1′,y1′, the left and right integral operators are defined as(12)ΔFκ,α,ξx1′+ϒ,γ,δ,k,ιΩx,σ;p=∫x1′xJxyEκ,α,ξγ,δ,k,ι,Δ;ϒΔ′yΩydy,(13)ΔFκ,α,ξy1′−ϒ,γ,δ,k,ιΩx,σ;p=∫xy1′JyxEκ,α,ξγ,δ,k,ι,Δ;ϒΔ′yΩydy,where(14)JxyEκ,α,ξγ,δ,k,ι,Δ;ϒ=ϒΔx−ΔyΔx−ΔyEκ,α,ξγ,δ,k,ισΔx−Δyκ;p.
Mittag–Leffler functions give several fractional integrals by assigning particular choice to the parameters involved in it, see Remarks 6 and 7 in [16].


## 3. Main Results

Throughout the paper, we use the following notation:(15)∫01huαh1−uαΔ′x−ux−x1′du=Hxx1′uα;h,Δ.Theorem 1.
LetΩ be a positive, refined α,h−m-convex and integrable function defined over x1′,y1′. Also, let ϒ/x be an increasing function defined on x1′,y1′ and Δ be strictly increasing and differentiable function on x1′,y1′. Then, for β,ξ,γ,ι∈ℝ,p,κ,ϑ,δ≥0,0<k≤δ+κ, and 0<k≤δ+ϑ, the following result holds:(16)ΔFk,β,ξ,y1′+ϒ,γ,δ,k,ιΩx,σ;p+ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΩx1′+mΩxmx−x1′Hxx1′uα;h,Δ+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩxmy1′−xHy1′xvα;h,Δ.Proof.
For the functionsϒ/x and Δ, the following inequality holds:(17)JxtEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′t≤Jxx1′Eκ,α,ξγ,δ,k,ι,Δ;ϒΔ′t.
Using refinedα,h−m-convexity of Ω, one can have(18)Ωt≤hx−tx−x1′αh1−x−tx−x1′αΩx1′+mΩxm.
From (17) and (18), we have the following integral inequality:(19)∫x1′xJxtEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′tΩtdt≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΩx1′+mΩxm×∫x1′xhx−tx−x1′αh1−x−tx−x1′αΔ′tdt.
Using (12) of Definition 10 on the left side of inequality (19) and making change of the variable by setting u=x−t/x−x1′ on the right-hand side of the above inequality, we obtain(20)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ωx1′+mΩxm×∫01huαh1−uαΔ′x−ux−x1′du.
Thus, we obtain(21)ΔFk,β,ξ,y1′+ϒ,γ,δ,k,ιΩx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ωx1′+mΩxmx−x1′Hxx1′uα;h,Δ.
Also, fort∈x,y1′ and x∈x1′,y1′, we can write(22)JtxEϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′t≤Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′t.
and(23)Ωt≤ht−xy1′−xαh1−t−xy1′−xαΩy1′+mΩxm.
From (22) and (23), we have the following integral inequality:(24)∫xy1′JtxEϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′tΩtdt≤Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩxm×∫xy1′ht−xy1′−xαh1−t−xy1′−xαΔ′tdt.
Using (13) of Definition 10 on the left-hand side and making change of the variable by setting v=t−x/y1′−x on the right-hand side of the above inequality, we obtain(25)ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx,σ;p≤Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩxmy1′−x×∫01hvαh1−vαΔ′x+vy1′−xdv.
Therefore,(26)ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx,σ;p≤Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩxmy1′−xHy1′xvα;h,Δ.
Combining (21) and (26), the required inequality (16) is obtained. Hence, the proof is completed.
Next, we give the refinement of Theorem1.Theorem 2.
Under the assumptions of Theorem1, if 0<ht<1, then the following result holds:(27)ΔFϑ,β,ξ,y1′+ϒ,γ,δ,k,ιΩx,σ;p+ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΩx1′+mΩxmx−x1′Hxx1′uα;h,Δ+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩxmy1′−xHy1′xvα;h,Δ≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ωx1′Hxx1′uα;h,Δ+mΩxmHxx1′1−uα;h,Δ+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒy1′−xΩy1′Hy1′xvα;h,Δ+mΩxmHy1′x1−vα;h,Δ.Proof.
From (18) and (23), one can see that, for 0<ht<1,(28)Ωt≤hx−tx−x1′αh1−x−tx−x1′αΩx1′+mΩxm≤hx−tx−x1′αΩx1′+mh1−x−tx−x1′αΩxm,Ωt≤ht−xy1′−xαh1−t−xy1′−xαΩy1′+mΩxm≤ht−xy1′−xαΩy1′+mh1−t−xy1′−xαΩxm.
Hence, by following the proof of Theorem1, one can obtain (27). Hence, the proof is completed.Corollary 1.
Under the assumptions of Theorem1, (16) gives the following result:(29)ΔFk,β,ξ,y1′+ϒ,γ,δ,k,ιΩx,σ;p+ΔFk,β,ξ,y1′−ϒ,γ,δ,k,ιΩx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ωx1′+mΩxmHxx1′uα;h,Δ+Jy1′xEκ,β,ξγ,δ,k,ι,Δ;ϒy1′−xΩy1′+mΩxmHy1′xvα;h,Δ.Now, we give the refinement of Theorem 5 in [9] in the following corollary.Corollary 2.
The following inequality for refinedh−m-convex function holds:(30)ΔFk,β,ξ,y1′+ϒ,γ,δ,k,ιΩx,σ;p+ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΩx1+mΩxmΔx−Δx1′+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩxmΔy1′−Δxh∞2≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΩx1′+mΩxmΔx−Δx1′+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩxmΔy1′−Δxh∞.Proof.
Usingα=1 and h∈L∞0,1 in (27), we obtain inequality (30).Remark 1.
(i)
Forϒx=xα′/kΓα′/kΓkα′, α′>k>0 with p=ω=0, inequality (16) coincides with Theorem 10 in [18](ii)
Fork=1 along with the conditions of (i), inequality (29) coincides with Theorem 6 in [18](iii)
ForΔ as identity function along with the conditions of (i), inequality (29) coincides with Theorem 5 in [14](iv)
ForΔ as identity function and k=1 along with the conditions of (i), inequality (29) coincides with Theorem 1 in [14](v)
Forht=t and m=1=α, inequality (16) coincides with Theorem 4 in [19](vi)
Forht=t and m=1=α, inequality (29) coincides with Corollary 1 in [19](vii)
Forht=t and m=1=α along with the conditions of (i), inequality (29) coincides with Theorem 3.1 in [20](viii)
Forht≤1/2 along with the conditions of (iv), inequality (29) coincides with Theorem 2 in [14](ix)
Forht=t and m=1=α along with the conditions of (iii), inequality (29) coincides with Corollary 8 in [14](x)
Forα=1 and ht=t along with the conditions of (iii), inequality (29) coincides with Corollary 14 in [14](xi)
Forht=ts and α=1 along with the conditions of (iii), inequality (29) coincides with Corollary 15 in [14](xii)
Forht=t and α=1 along with the conditions of (iii), inequality (29) coincides with Corollary 16 in [14](xiii)
Forht=t and m=1=α along with the conditions of (iv), inequality (29) coincides with Corollary 1 in [14](xiv)
Forα=1 and ht=t along with the conditions of (iv), inequality (29) coincides with Corollary 2 in [14](xv)
Forht=ts and α=1 along with the conditions of (iv), inequality (29) coincides with Corollary 4 in [14](xvi)
Forht=t and α=1 along with the conditions of (iv), inequality (29) coincides with Corollary 5 in [14]
By using0<ht<1 and making different choices of functions h and Δ and the parameters in (16), one can get the refinements of many well-known inequalities for different classes of convex functions which are mentioned in Remark 3 in [9].
Next, we give a lemma which we will use in the proof of upcoming Theorem3.Lemma 1.
LetΩ:0,∞⟶ℝ be a refined α,h−m-convex function. If Ωx=Ωx1′+y1′−x/m,x∈x1′,y1′, and m∈0,1, then the following inequality holds:(31)Ωx1′+y1′2≤h12αh2α−12αm+1Ωx.Proof.
SinceΩ is refined α,h−m-convex, then following inequality holds:(32)Ωx1′+y1′2≤h12αh2α−12α×Ωx−x1′y1′−x1′y1′+y1′−xy1′−x1′x1′+mΩx−x1′/y1′−x1′x1′+y1′−x/y1′−x1′y1′m≤h12αh2α−12αΩx+mΩx1′+y1′−xm.
UsingΩx=Ωx1′+y1′−x/m in the above inequality, we obtain (31). This completes the proof.Remark 2.
(i)
Forht=t and m=α=1, (31) coincides with Lemma 1 in [19](ii)
For0<ht<1, (31) gives refinement of Lemma 1 in [9]Theorem 3.
Under the assumptions of Theorem1, the following result holds for Ωx=Ωx1′+y1′−x/m:(33)1h1/2αh2α−1/2αm+1Ωx1′+y1′2×ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ι1x1′,σ;p+ΔFϑ,β,ξ,y1′+ϒ,γ,δ,k,ι1y1′,σ;p≤ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx1′,σ;p+ΔFϑ,β,ξ,y1′+ϒ,γ,δ,k,ιΩy1′,σ;p≤y1′−x1′Ωy1′+mΩx1′mJy1′x1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒHy1′x1′vα;h,Δ+Jy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒHy1′x1′vα;h,Δ.Proof.
For the kernel defined in (14) and function Δ, the following inequality holds:(34)Jxx1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′x≤Jy1′x1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′x,x∈x1′,y1′.
Using refinedα,h−m-convexity of Ω, we have(35)Ωx≤hx−x1′y1′−x1′αh1−x−x1′y1′−x1′αΩy1′+mΩx1′m.
From (34) and (35), we have the following integral inequality:(36)∫x1′y1′Jxx1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒΩxΔ′xdx≤Jy1′x1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩx1′m×∫x1y1′hx−x1′y1′−x1′αh1−x−x1′y1′−x1′αΔ′xdx.
Using (13) of Definition 10 on the right-hand side and making change of the variable by setting v=x−x1′/y1′−x1′ on the right-hand side of the above inequality, we obtain(37)ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ι1Ωx1′,σ;p≤Jy1′x1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒy1′−x1′Ωy1′+mΩx1′mHy1′x1′vα;h,Δ.
The following inequality also holds true forx∈x1′,y1′:(38)Jy1′xEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′x≤Jy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΔ′x.
From (35) and (38), the following integral inequality is obtained:(39)∫x1′y1′Jy1′xEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′xΩxdx≤Jy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΩy1′+mΩx1m×∫x1′y1′hx−x1′y1′−x1′αh1−x−x1′y1′−x1′αΔ′xdx.
Using (12) of Definition 10 on the left-hand side and making change of the variable on the right-hand side of the above inequality, we obtain(40)ΔFk,β,ξ,y1′+ϒ,γ,δ,k,ι1Ωy1′,σ;p≤Jy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒy1′−x1′Ωy1′+mΩx1′mHy1′x1′vα;h,Δ.
Now, using Lemma1, we can write(41)∫x1′y1′Ωx1′+y1′2Jxx1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′xdx≤h12αh2α−12αm+1∫x1′y1′Jxx1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′xΩxdx,which by using (13) of Definition 10 gives the following integral inequality:(42)1h1/2αh2α−1/2αm+1Ωx1′+y1′2ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ι1x1′,σ;p≤ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx1′,σ;p.
Again, using Lemma1, we can write(43)Ωx1′+y1′2Jy1′xEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′xdx≤h12αh2α−12αm+1∫x1′y1′Jy1′xEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′xΩxdx,which by using (12) of Definition 10 gives the following fractional integral inequality:(44)1h1/2αh2α−1/2αm+1Ωx1′+y1′2ΔFϑ,β,ξ,y1′+ϒ,γ,δ,k,ι1y1′,σ;p≤ΔFϑ,β,ξ,y1′+ϒ,γ,δ,k,ιΩy1′,σ;p.
Inequality (33) will be obtained by using (37), (40), (42), and (44).
The following theorem is the refinement of Theorem3.Theorem 4.
Under the assumptions of Theorem3, if 0<ht<1, then the following refinement holds:(45)1h1/2αh2α−1/2αm+1Ωx1′+y1′2×ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ι1x1′,σ;p+ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ι1y1′,σ;p≤1h1/2α+mh2α−1/2αΩx1′+y1′2×ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ι1x1′,σ;p+ΔFκ,β,ξ,x1′+ϒ,γ,δ,k,ι1y1′,σ;p≤ΔFκ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx1′,σ;p+ΔFϑ,β,ξ,x1′+ϒ,γ,δ,k,ιΩy1′,σ;p≤y1′−x1′Ωy1′+mΩx1′mJy1′x1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒHy1′x1′vα;h,Δ+Jy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒHy1′x1′vα;h,Δ≤y1′−x1′Jy1′x1′Eϑ,β,ξγ,δ,k,ι,Δ;ϒ+Jy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒ×Ωy1′Hy1′x1′vα;h,Δ+mΩx1′mHy1′x1′1−vα;h,Δ.Proof.
From (35), one can see that, for 0<ht<1,(46)Ωx≤hx−x1′y1′−x1′αh1−x−x1′y1′−x1′αΩy1′+mΩx1′m≤hx−x1′y1′−x1′αΩy1′+mh1−x−x1′y1′−x1′αΩx1′m.
Hence, by following the proof of Theorem3, one can obtain (45). This completes the proof.Corollary 3.
Under the assumptions of Theorem3, (33) gives the following result:(47)1h1/2αh2α−1/2αm+1Ωx1′+y1′2ΔFκ,β,ξ,y1′−ϒ,γ,δ,k,ι1x1′,σ;p+ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ι1y1′,σ;p≤ΔFκ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx1′,σ;p+ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩy1′,σ;p≤2y1′−x1′Ωy1′+mΩx1′mJy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒHy1′x1′vα;h,Δ.
Now, we give the refinement of Theorem 6 in [9] in the following corollary.Corollary 4.
The following inequality for refinedh−m-convex function holds:(48)1h21/2m+1Ωx1′+y1′2ΔFκ,α,ξ,y1′−ϒ,γ,δ,k,ι1x1′,σ;p+ΔFκ,α,ξ,x1′+ϒ,γ,δ,k,ι1y1′,σ;p≤1h1/2m+1Ωx1′+y1′2ΔFκ,α,ξ,y1′−ϒ,γ,δ,k,ιΩx1′,σ;p+ΔFκ,α,ξ,x1′+ϒ,γ,δ,k,ι1y1′,σ;p≤ΔFκ,β,ξ,y1′−ϒ,γ,δ,k,ιΩx1′,σ;p+ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩy1′,σ;p≤2Ωy1′+mΩx1′mJy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΔy1′−Δx1′h∞2≤2Ωy1′+mΩx1′mJy1′x1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΔy1′−Δx1′h∞.Proof.
Forh∈L∞0,1 and α=1 in (45), one can obtain (48).Remark 3.
(i)
Forht=t and m=1=α, inequality (33) coincides with Theorem 5 in [19](ii)
Forht=t and m=1=α, inequality (47) coincides with Corollary 2 in [19]
By using0<ht<1 and making different choices of functions h and Δ and the parameters in (33), one can get the refinements of many well-known inequalities for different classes of convex functions which are mentioned in Remark 5 of [9].Theorem 5.
LetΩ,Δ be differentiable functions such that Ω′ is refined α,h−m-convex and Δ be strictly increasing over x1′,y1′ and differentiable over x1′,y1′. Also, ϒ/x be an increasing function on x1′,y1′ and β,ξ,γ,ι∈ℝ,p,κ,ϑ,δ≥0, 0<k≤δ+κ, and 0<k≤δ+ϑ. Then, for x∈x1′,y1′, we have(49)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩ∗Δx,σ;p+ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩ∗Δx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ω′x1′+mΩ′xmHxx1′uα;h,Δ+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒy1′−xΩ′y1′+mΩ′xmHy1′xvα;h,Δ,where(50)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩ∗Δx,σ;p=∫x1′xJxtEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′tΩ′tdt,ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩ∗Δx,σ;p=∫xy1′JtxEϑ,β,ξγ,δ,k,ι,Δ;ϒΔ′tΩ′tdt.Proof.
Using refinedα,h−m-convexity of Ω′ over x1′,y1′ implies(51)Ω′t≤hx−tx−x1′αh1−x−tx−x1′αΩ′x1′+mΩ′xm.
Absolute value property implies the following relation:(52)−hx−tx−x1′αh1−x−tx−x1′αΩ′x1′+mΩ′xm≤Ω′t≤hx−tx−x1′αh1−x−tx−x1′αΩ′x1′+mΩ′xm.
From (17) and the second inequality of (52), we have the following inequality:(53)∫x1xJxtEκ,β,ξγ,δ,k,ι,Δ;ϒΔ′tΩ′tdt≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΩ′x1′+mΩ′xm×∫x1′xhx−tx−x1′αh1−x−tx−x1′αΔ′tdt,which leads to the following fractional integral inequality:(54)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩ∗Δx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ω′x1′+mΩxmHxx1′uα;h,Δ.
Also, inequality (17) and the first inequality of (52) give the following fractional integral inequality:(55)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩ∗Δx,σ;p≥−Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ω′x1′+mΩ′xmHxx1′uα;h,Δ.
Again, using refinedα,h−m-convexity of Ω′ over x1′,y1′, we can write(56)Ω′t≤ht−xy1′−xαh1−t−xy1′−xαmΩ′xm+Ω′y1′.
and(57)ht−xy1′−xαh1−t−xy1′−xαmΩ′xm+Ω′y1′≤Ω′t≤ht−xy1′−xαh1−t−xy1′−xαmΩ′xm+Ω′y1′.
From (22) and the second inequality of (57), the following fractional integral inequality is obtained:(58)ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩ∗Δx,σ;p≤Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒy1′−xΩ′y1′+mΩ′xmHy1′xvα;h,Δ,and (22) and the first inequality of (57) give the following fractional integral inequality:(59)ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ιΩ∗Δx,σ;p≥−Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒy1′−xΩ′y1′+mΩ′xmHy1′xvα;h,Δ.
Inequality (49) will be obtained by using (54), (55), (58), and (59). Hence, the proof is completed.
Next, we give refinement of Theorem5 in the following theorem.Theorem 6.
Under the assumptions of Theorem5, if 0<ht<1, then the following refinement holds:(60)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ι,Ω∗Δx,σ;p+ΔFϑ,β,ξ,y1′−ϒ,Ω∗γ,δ,k,ι,Δx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ω′x1′+mΩ′xmHxx1′uα;h,Δ+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒy1′−xΩ′y1′+mΩ′xmHy1′xvα;h,Δ≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ω′x1′Hxx1′uα;h,Δ+mΩ′xmHxx1′1−uα;h,Δ+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒy1′−x×Ω′y1′Hy1′xvα;h,Δ+mΩ′xmHy1′x1−vα;h,Δ.Proof.
From (51), one can see that, for 0<ht<1,(61)Ω′t≤hx−tx−x1′αh1−x−tx−x1′αΩ′x1′+mΩ′xm≤hx−tx−x1′αΩ′x1′+mh1−x−tx−x1′αΩ′xm.
Hence, by following the proof of Theorem5, one can obtain (60). This completes the proof.Corollary 5.
Under the assumptions of Theorem5, (49) gives the following result:(62)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩ∗Δx,σ;p+ΔFκ,β,ξ,y1′−ϒ,γ,δ,k,ιΩ∗Δx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒx−x1′Ω′x1′+mΩ′xmHxx1′uα;h,Δ+Jy1′xEκ,β,ξγ,δ,k,ι,Δ;ϒy1′−xΩ′y1′+mΩ′xmHy1′xvα;h,Δ.The following corollary presents the refinement of Theorem 7 in [9].Corollary 6.
The following inequality for refinedh−m-convex function holds:(63)ΔFκ,β,ξ,y1′+ϒ,γ,δ,k,ιΩ∗Δx,σ;p+ΔFϑ,β,ξ,y1′−ϒ,γ,δ,k,ι1Ω∗gx,σ;p≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΔx−Δx1′Ω′x1′+mΩ′xm+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΔy1′−ΔxΩ′y1′+mΩ′xmh∞2≤Jxx1′Eκ,β,ξγ,δ,k,ι,Δ;ϒΔx−Δx1′Ω′x1′+mΩ′xm+Jy1′xEϑ,β,ξγ,δ,k,ι,Δ;ϒΔy1′−ΔxΩ′y1′+mΩ′xmh∞.Proof.
Forh∈L∞0,1 and α=1 in (60), we obtain (63).Remark 4.
(i)
Forht=t and m=1=α, inequality (49) coincides with Theorem 6 in [19](ii)
Forht=t and m=1=α, inequality (62) coincides with Corollary 3 in [19]
By using0<ht<1 and making different choices of functions h and Δ and the parameters in (49), one can get the refinements of many well-known inequalities for different classes of convex functions which are mentioned in Remark 6 of [9].


## 4. Inequalities for Fractional Integral Operators

In this section, we present the bounds of some of the fractional integral operators which will be deduced from the results of Section3.Proposition 1.
Under the assumptions of Theorem1, the following result holds:(64)ΓβΔβIx1′+Ωx+ΔβIy1′−Ωx≤Δx−Δx1′β−1Ωx1′+mΩxmx−x1′Hxx1′uα;h,Δ+Δy1′−Δxβ−1Ωy1′+mΩxmy1′−xHy1′xvα;h,Δ.Proof.
Forϒt=tβ,β>0, and p=σ=0 in the proof of Theorem 1, we obtain (64).Proposition 2.
Under the assumptions of Theorem1, the following inequality holds:(65)Γβx1′+ΙϒΩx+y1′+ΙϒΩx≤ϒx−x1′Ωx1′+mΩxm∫01huαh1−uαdu+ϒy1′−xΩy1′+mΩxm∫01hvαh1−vαdv.Proof.
UsingΔ as identity function with σ=p=0 in the proof of Theorem 1, we obtain the required result.Corollary 7.
Forϒt=Γβtβ/k/kΓkβ with β>k and p=σ=0, (12) and (13) reduce to the fractional integral operators (8) and (9), which satisfy the following upper bound:(66)kΓkβΔβIx1′+κΩx+ΔβIy1′+κΩx≤Δx−Δx1′β/k−1Ωx1′+mΩxmx−x1′Hxx1′uα;h,Δ+Δy1′−Δxβ/k−1Ωy1′+mΩxmy1′−xHy1′xvα;h,Δ.Remark 5.
For0<ht<1, (66) gives refinement of Corollary 8 in [9].Corollary 8.
Usingϒt=tβ and Δ as identity function for β≥1 along with p=σ=0, (12) and (13) give fractional integral Iβx1′+Ωx and Iβy1′−Ωx defined in [15], which satisfy the following upper bound:(67)ΓβΔβIx1′+Ωxx+ΔβIy1′+Ωx≤x−x1′βΩx1′+mΩxm∫01huαh1−uαdu+y1′−xβΩy1′+mΩxm∫01hvαh1−vαdv.Corollary 9.
Usingϒt=Γβtβ/k/kΓkβ and Δ as identity function along with p=σ=0, (12) and (13) reduce to the fractional integral operators βIx1′+κΩx and βIy1′+κΩx given in [21], which satisfy the following upper bound:(68)βIx1′+κΩx+βIy1′+κΩx≤1kΓkβx−x1′β/kΩx1′+mΩxm×∫01huαh1−uαdu+y1′−xβ/kΩy1′+mΩxm×∫01hvαh1−vαdv.Remark 6.
For0<ht<1, (68) gives refinement of Corollary 10 in [9].Corollary 10.
Fork=1 in Corollary 9, the following upper bound for Riemann–Liouville fractional integral is satisfied:(69)βIx1′+Ωx+βIy1′+Ωx≤1Γβx−x1′βΩx1′+mΩxm×∫01huαh1−uαdu+y1′−xβΩy1′+mΩxm×∫01hvαh1−vαdv.Remark 7.
For0<ht<1, (69) gives refinement of Corollary 11 in [9].
Similar bounds can be obtained for Theorems3 and 5, which we leave for the reader.


## 5. Conclusions

This article is about the bounds of unified integral operators via refinedα,h−m-convexity. The obtained results are the refinements of some already published results. Moreover, some deducible fractional integral operators and their related bounds are also given.



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