# Inequalities for a Unified Integral Operator for Stronglyα,m-Convex Function and Related Results in Fractional Calculus

Authors: Chahn Yong Jung; Ghulam Farid; Kahkashan Mahreen; Soo Hak Shim
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we study integral inequalities which will provide refinements of bounds of unified integral operators established for convex andα,m-convex functions. A new definition of function, namely, strongly α,m-convex function is applied in different forms and an extended Mittag-Leffler function is utilized to get the required results. Moreover, the obtained results in special cases give refinements of fractional integral inequalities published in this decade.


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## Body

## 1. Introduction

Fractional integral operators are very useful and are extensively utilized in mathematics, physics, engineering, and many other subjects. The researchers have introduced variety of fractional integral operators, most of them generalize classical Riemann-Liouville integrals. In recent study of mathematical inequalities, fractional integral operators are playing an important role. Many classical inequalities have been studied for fractional integral operators of different kinds. For example, one can see recent articles dealing with fractional inequalities in [1–15] and in the references therein. Our aim in this article is to study an integral operator for a newly defined strongly α,m-convex function, which has direct consequences in fractional integral operators. Next, we give some definitions of generalized fractional integral operators.Definition 1 (see [16]).
Letf:a,b→ℝ be an integrable function. Also let g be an increasing and positive function on a,b, having a continuous derivative g on a,b. The left-sided and right-sided fractional integrals of a function f with respect to another function g on a,b of order μ where Rμ>0 are defined by(1)gμIa+fx=1Γμ∫axgx−gtμ−1gtftdt,x>a,gμIb−fx=1Γμ∫xbgt−gxμ−1gtftdt,x<b,where Γ. is the gamma function.Ak-analogue of the above definition is defined as follows.Definition 2 (see [17]).
Letf:a,b→ℝ be an integrable function. Also let g be an increasing and positive function on a,b, having a continuous derivative g on a,b. The left-sided and right-sided fractional integrals of a function f with respect to another function g on a,b of order μ;Rμ,k>0 are defined by(2)gμIa+kfx=1kΓkμ∫axgx−gtμ/k−1gtftdt,x>a,gμIb−kfx=1kΓkμ∫xbgt−gxμ/k−1gtftdt,x<b,where Γk. is given by(3)Γkx=∫0∞tx−1e−tk/kdt,Rx>0.A well-known function named Mittag-Leffler function is defined by [18](4)Eαz=∑n=0∞znΓαn+1,where α,z∈ℂ and Rα>0.One can see the references [19–22] to study the Mittag-Leffler function and its generalizations.Fractional integral operator containing an extended Mittag-Leffler function is defined as follows.Definition 3 (see [1]).
Letω,μ,α′,l,γ,c∈ℂ, Rμ,Rα,Rl>0, Rc>Rγ>0 with p≥0, δ>0 and 0<k≤δ+Rμ. Let f∈L1a,b and x∈a,b. Then, the generalized fractional integral operators εμ,α′,l,ω,a+γ,δ,k,cf and εμ,α′,l,ω,b−γ,δ,k,cf are defined by(5)εμ,α′,l,ω,a+γ,δ,k,cfx;p=∫axx−tα′−1Eμ,α′,lγ,δ,k,cωx−tμ;pftdt,εμ,α′,l,ω,b−γ,δ,k,cfx;p=∫xbt−xα′−1Eμ,α′,lγ,δ,k,cωt−xμ;pftdt,where(6)Eμ,α′,lγ,δ,k,ct;p=∑n=0∞βpγ+nk,c−γβγ,c−γcnkΓμn+α′tnlnδ,is the extended generalized Mittag-Leffler function.Recently, Farid defined the following unified integral operator which unifies several kinds of fractional integrals. Also, new kinds of fractional integrals can be generated from it.Definition 4 (see [23]).
Letf,g:a,b→ℝ, 0<a<b, be the functions such thatfbe positive andf∈L1a,b, andgbe differentiable and strictly increasing. Also letϕ/xbe an increasing function ona,∞andα′,l,γ,c∈ℂ, Rα′,Rl>0, Rc>Rγ>0, p,μ,δ≥0and0<k≤δ+μ. Then, forx∈a,b, the left and right integral operators are defined by(7)gFμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p=∫axKxyEμ,α′,lγ,δ,k,c,g;ϕfydgy,(8)gFμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p=∫xbKyxEμ,α′,lγ,δ,k,c,g;ϕfydgy,where KxyEμ,α′,lγ,δ,k,c,g;ϕ=ϕgx−gy/gx−gyEμ,α′,lγ,δ,k,cωgx−gyμ;p.The following property of the kernel used in the unified integral operator will be used in sequel.P: Let g and ϕ/I be increasing functions. Then, for m<t<n, m,n∈a,b, the kernel KmnEμ,α′,lγ,δ,k,c,g;ϕ satisfies the following inequality:(9)KtmEμ,α′,lγ,δ,k,c,g;ϕg′t≤KnmEμ,α′,lγ,δ,k,c,g;ϕg′t.This can be obtained from following two straightforward inequalities:(10)ϕgt−gmgt−gmg′t≤ϕgn−gmgn−gmg′t,Eμ,α′,lγ,δ,k,cωgt−gmμ;p≤Eμ,α′,lγ,δ,k,cωgn−gmμ;p.The reverse of inequality (9) holds when g and ϕ/I are of opposite monotonicity. For suitable settings of functions ϕ, g and certain values of parameters included in the Mittag-Leffler function (6), many well-known fractional integral operators can be reproduced, see ([11], Remarks 6 and 7).The objective of this paper is to obtain bounds of unified integral operators using stronglyα,m-convexity. It is defined as follows [24].Definition 5.
A functionf:0,b→ℝ,b>0 is said to be strongly α,m-convex, where α,m∈0,12 if(11)ftx+m1−ty≤tαfx+m1−tαfy−λmtα1−tαy−x2,holds for all x,y∈0,b and t∈0,1.Remark 6.
(i) If we putλ=0, then (11) gives the definition of α,m-convex functions
(ii) If we putα=1, then (11) gives the definition of strongly m-convex functions
(iii) If we putα,m=1,m, then (11) gives the definition of m-convex function
(iv) If we putα,m=1,1, then (11) gives the definition of convex function
(v) If we putα,m=1,0, then (11) gives the definition of star-shaped function
(vi) If we putλ=0 and α,m=1,0, then (11) gives the definition of convex functionIn the upcoming section, bounds of unified integral operators are established by using stronglyα,m-convexity. These bounds provide general formulas to get bounds of many fractional integral operators along with described in [11], Remarks 6 and 7]. Many mathematicians worked on new types of Hadamard inequalities using convex functions, see [9, 25–27]. We also established general Hadamard type inequality by applying Lemma 10 which further produces various inequalities of Hadamard type for fractional integrals.


## 2. Main Results

Theorem 7.
Letf:a,b→ℝ be a positive integrable strongly α,m-convex function with m∈0,1, 0<a<mb. Let g:a,b→ℝ be differentiable and strictly increasing function, also let ϕ/x be an increasing function on a,b. Then, unified integral operators (7) and (8) satisfy the following inequality:(12)gFμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p+gFμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕmfxmgx−faga−Γα+1x−aαmfxm−faαIa+gx−λx−ma2mx−aαΓα+1aIa+gx−Γ2α+12αIa+gxx−aα+KbxEμ,β,lγ,δ,k,c,g;ϕfbgb−mfxmgx−Γα+1b−xαfb−mfxmαIb−gx−λmb−x2mb−xαΓ2α+12αIb−gxb−xα−Γα+1αIb−gx,where αIa+gx,αIb−gx are Riemann-Liouville fractional integrals.Proof.
According to propertyP, the following inequalities hold:(13)KxtEμ,α′,lγ,δ,k,c,g;ϕg′t≤KxaEμ,α′,lγ,δ,k,c,g;ϕg′t,t∈a,x,(14)KxtEμ,α′,lγ,δ,k,c,g;ϕg′t≤KbxEμ,β,lγ,δ,k,c,g;ϕg′t,t∈x,b.Stronglyα,m-convex function f satisfies the following inequalities:(15)ft≤x−tx−aαfa+m1−x−tx−aαfxm−λmx−tx−aα1−x−tx−aαx−mam2,(16)ft≤t−xb−xαfb+m1−t−xb−xαfxm−λmt−xb−xα1−t−xb−xαmb−xm2.Multiplying (13) with (15) and integrating over a,x, one can obtain(17)∫axKxtEμ,α′,lγ,δ,k,c,g;ϕftdgt≤faKxaEμ,α′,lγ,δ,k,c,g;ϕ∫axx−tx−aαdgt+mfxmKxaEμ,α′,lγ,δ,k,c,g;ϕ∫ax1−x−tx−aαdgt−λmx−ma2KxaEμ,α′,lγ,δ,k,c,g;ϕ∫axx−tx−aα1−x−tx−aαdgt.From the above inequality, one can obtain(18)gFμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕmfxmgx−faga−Γα+1x−aαmfxm−faαIa+gx−λx−ma2mx−aαΓα+1αIa+gx−Γ2α+12αIa+gxx−aα.Now, adopting the same procedure as we did for (13) and (15), the following inequality can be obtained from (14) and (16):(19)gFμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KbxEμ,β,lγ,δ,k,c,g;ϕfbgb−mfxmgx−Γα+1b−xαfb−mfxmαIb−gx−λmb−x2mb−xαΓ2α+12αIb−gxb−xα−Γα+1αIb−gx.From inequalities (18) and (19), inequality (12) can be obtained.Corollary 8.
If we considerα=1 and λ=0 in (12), then the following inequality holds for m-convex functions:(20)gFμ,α′,l,a+ϕ,γ,δ,k,cfx,ω;p+gFμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕgx−gamfxm+fa+KbxEμ,β,lγ,δ,k,c,g;ϕgb−gxfb+mfxm.Remark 9.
(i) If we considerλ=0 in (12), ([28], Theorem 7) is obtained, otherwise the refinement is obtained
(ii) If we considerα,m=1,1 in (12), ([12], Theorem 4]) is obtained
(iii) If we considerϕt=tα′ and gx=x, in (12), then ([24], Theorem 4]) is obtained
(iv) If we considerλ=0, α,m=1,1 in (12), ([11], Theorem 8]) is obtained
(v) If we considerλ=0, ϕt=Γα′tα′/k/kΓkα′ and p=ω=0 in (12), then ([10], Theorem 7]) can be obtained
(vi) If we considerα′=β in the result of (v), then ([10], Corollary 8]) can be obtained
(vii) If we considerλ=0, ϕt=Γμtμ, p=ω=0, and α,m=1,1 in (12), ([6], Theorem 7]) is obtained
(viii) If we considerα′=β in the result of (vi), ([6], Corollary 8]) is obtained
(ix) If we considerλ=0, ϕt=Γα′tα′/k/kΓkα′, α,m=1,1, gx=x, and p=ω=0, then ([4], Theorem 7]) can be obtained
(x) If we considerα′=β in the result of (xi), then ([4], Corollary 8]) can be obtained
(xi) If we considerλ=0, ϕt=Γα′tα′, gx=x, and p=ω=0 and α,m=1,1 in (12), then ([5], Theorem 7]) is obtained
(xii) If we considerα′=β in the result of (xi), ([5], Corollary 8]) can be obtained
(xiii) If we considerα′=β=1 and x=a or x=b in the result of (xii), ([5], Corollary 12]) can be obtained
(xiv) If we considerα′=β=1 and x=a+b/2 in the result of (xii), ([5], Corollary 15]) can be obtainedTo prove the next result, we need the following lemma.Lemma 10 (see [24]).
Letf:a,b→ℝbe stronglyα,m-convex function,0<a<mb. Iffa+mb−x/m=fxandα,m∈0,12withm∈0,1, then the following inequality holds:(21)fa+mb2≤fx12α+m1−12α−λm2α−122α×a+mb−x−mx2.The following result provides upper and lower bounds of sum of operators (7) and (8) in the form of a Hadamard inequality.Theorem 11.
With the assumptions of Theorem7 in addition if fx=fa+mb−x/m with m∈0,1, then we have(22)2α1+m2α−1fa+mb2gFμ,β,l,b−ϕ,γ,δ,k,c1a,ω;p+gFμ,α′,l,a+ϕ,γ,δ,k,c1b,ω;p+λm2α−122αgFμ,β,l,b−ϕ,γ,δ,k,ca+mb−x−mx2a,ω;p+gFμ,α′,l,a+ϕ,γ,δ,k,ca+mb−x−mx2b,ω;p≤gFμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p+gFμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p≤KbaEμ,α′,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕ×fbgb−mfamga−Γα+1b−aαfb−mfamαIb−ga−λmb−a2mb−aαΓ2α+12αIb−gab−aα−Γα+1αIb−ga.Proof.
According toP, the following inequalities hold:(23)KxaEμ,α′,lγ,δ,k,c,g;ϕg′x≤KbaEμ,α′,lγ,δ,k,c,g;ϕg′x,(24)KxaEμ,β,lγ,δ,k,c,g;ϕg′x≤KbaEμ,β,lγ,δ,k,c,g;ϕg′x.
Using the definition of stronglyα,m-convex function f, the following inequality holds:(25)fx≤x−ab−aαfb+m1−x−ab−aαfam.Multiplying (23) and (25) and integrating the resulting inequality over a,b, one can obtain(26)∫abKxaEμ,α′,lγ,δ,k,c,g;ϕfxdgx≤mfamKbaEμ,α′,lγ,δ,k,c,g;ϕ∫ab1−x−ab−aαdgx+fbKbaEμ,α′,lγ,δ,k,c,g;ϕ∫abx−ab−aαdgx−λmx−ma2KbaEμ,α′,lγ,δ,k,c,g;ϕ∫abx−ab−aα1−x−ab−aαdgx.From the above inequality, one can obtain(27)gFμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p≤KbaEμ,α′,lγ,δ,k,c,g;ϕfbgb−mfamga−Γα+1b−aαfb−mfamαIb−ga−λmb−a2mb−aαΓ2α+12αIb−gab−aα−Γα+1αIb−ga.Adopting the same pattern of simplification as we did for (23) and (25), the following inequality can be obtained from (25) and (24)(28)gFμ,β,l,b−ϕ,γ,δ,k,cfa;p≤KbaEμ,β,lγ,δ,k,c,g;ϕfbgb−mfamga−Γα+1b−aαfb−mfamαIb−ga−λmb−a2mb−aαΓ2α+12αIb−gab−aα−Γα+1αIb−ga.From (27) and (28), the following inequality can be obtained:(29)gFμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p+gFμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p≤fbgb−mfamga−Γα+1b−aαfb−mfamαIb−ga−λmb−a2mb−aαΓ2α+12αIb−gab−aα−Γα+1αIb−gaKbaEμ,α′,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕ.Multiplying both sides of (21) by KxaEμ,β,lγ,δ,k,c,g;ϕdgx, and integrating over a,b, we have(30)fa+mb2∫abKxaEμ,β,lγ,δ,k,c,g;ϕdgx≤12α+m1−12α×∫abKxaEμ,β,lγ,δ,k,c,g;ϕfxdgx−λm2α−122α×∫abKxaEμ,β,lγ,δ,k,c,g;ϕa+mb−x−mx2dgx.From the above inequality, one can obtain(31)2α1+m2α−1fa+mb2gFμ,β,l,b−ϕ,γ,δ,k,c1a,ω;p+λm2α−122α×gFμ,β,l,b−ϕ,γ,δ,k,ca+mb−x−mx2a,ω;p≤gFμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p.Similarly multiplying both sides of (21) by KbxEμ,α′,lγ,δ,k,c,g;ϕdgx, and integrating over a,b, we have(32)2α1+m2α−1fa+mb2gFμ,α′,l,b−ϕ,γ,δ,k,c1b,ω;p+λm2α−122α×gFμ,α′,l,a+ϕ,γ,δ,k,ca+mb−x−mx2b,ω;p≤gFμ,α′,l,a+ϕ,γ,δ,k,cfb,ω;p.From inequalities (29), (31), and (32), inequality (22) can be obtained.Corollary 12.
If we considerα=1 and λ=0 in (22), then the following inequality holds for m-convex function:(33)21+mfa+mb2gFμ,β,l,b−ϕ,γ,δ,k,c1a,ω;p+gFμ,α′,l,a+ϕ,γ,δ,k,c1b,ω;p≤gFμ,α′,l,b−ϕ,γ,δ,k,cfa,ω;p+gFμ,β,l,a+ϕ,γ,δ,k,cfb,ω;p≤KbaEμ,α′,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕgb−gafb+mfam.Remark 13.
(i) If we considerα,m=1,1 in (22), ([29], Theorem 5]) is obtained
(ii) If we considerϕt=tα′ and gx=x, in (22), then ([24], Theorem 6]) is obtained
(iii) If we considerλ=0, α,m=1,1 in (22), ([11], Theorem 22]) is obtained
(iv) If we consider andϕt=Γα′tα′+1 in (22), p=ω=0, and α,m=1,1 in (22), ([6], Theorem 14]) is obtained
(v) If we considerα′=β in the result of (iv), ([6], Corollary 15]) is obtained
(vi) If we considerλ=0, ϕt=Γα′tα′/k+1, α,m =(1,1), gx=x and p=ω=0 in (22), then ([4], Theorem 14]) can be obtained
(vii) If we considerα′=β in the result of (vi), then ([4], Corollary 6]) can be obtained
(viii) If we considerλ=0, ϕt=Γα′tα′+1p=ω=0, α,m=1, and gt=t in (22), ([5], Theorem 14]) can be obtained
(ix) If we considerα′=β in the result of (viii), ([5], Corollary 6]) can be obtainedTheorem 14.
Letf:a,b→ℝ be a differentiable function. If ∣f′∣ is strongly α,m-convex with m∈0,1,0<a<mb and g:a,b→ℝ be differentiable and strictly increasing function, also let ϕ/x be an increasing function on a,b. Then, for unified integral operators, the following inequality holds:(34)gFμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p+gFμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕmf′xmgx−f′aga−Γα+1x−aαmf′xm−f′aαIa+gx−λx−ma2mx−aαΓα+1αIa+gx−Γ2α+12αIa+gxx−aα+KbxEμ,β,lγ,δ,k,c,g;ϕfbgb−mfxmgx−Γα+1b−xαfb−mfxmαIb−gx−λmb−x2mb−xαΓ2α+12αIb−gxb−xα−Γα+1αIb−gx,where(35)gFμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p≔∫axKxtEμ,α′,lγ,δ,k,c,g;ϕf′tdgt,gFμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≔∫xbKtxEμ,α′,lγ,δ,k,c,g;ϕf′tdgt.Proof.
Using the definition of stronglyα,m-convexity for ∣f′∣, the following inequality is valid:(36)∣f′t∣≤x−tx−aα∣f′a∣+m1−x−tx−aαf′xm−λmx−tx−aα1−x−tx−aαx−mam2.The inequality (36) can be written as follows:(37)−x−tx−aα∣f′a∣+m1−x−tx−aαf′xm−λmx−tx−aα1−x−tx−aαx−mam2≤f′t≤x−tx−aα∣f′a∣+m1−x−tx−aαf′xm−λmx−tx−aα1−x−tx−aαx−mam2.We consider the second inequality of (37)(38)f′t≤x−tx−aα∣f′a∣+m1−x−tx−aαf′xm−λmx−tx−aα1−x−tx−aαx−mam2.Multiplying (13) and (38) and integrating over a,x, we can obtain(39)∫axKxtEμ,α′,lγ,δ,k,c,g;ϕf′tdgt≤∣f′a∣KxaEμ,α′,lγ,δ,k,c,g;ϕ∫axx−tx−aαdgt+mf′xmKxaEμ,α′,lγ,δ,k,c,g;ϕ∫ax1−x−tx−aαdgt−λmx−ma2KxaEμ,α′,lγ,δ,k,c,g;ϕ∫axx−tx−aα1−x−tx−aαdgt.From the above inequality, one can obtain(40)gFμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕmf′xmgx−∣f′a∣ga−Γα+1x−aαmf′xm−∣f′a∣αIa+gx−λx−ma2mx−aαΓα+1αIa+gx−Γ2α+12αIa+gxx−aα.If we consider the left hand side from inequality (37) and adopt the same pattern as did for the right hand side inequality, then(41)gFμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p≥−KxaEμ,α′,lγ,δ,k,c,g;ϕmf′xmgx−∣f′a∣ga−Γα+1x−aαmf′xm−∣f′a∣αIa+gx−λx−ma2mx−aαΓα+1αIa+gx−Γ2α+12αIa+gxx−aα.From (40) and (41), the following inequality is obtained:(42)gFμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕmf′xmgx−∣f′a∣ga−Γα+1x−aαmf′xm−∣f′a∣αIa+gx−λx−ma2mx−aαΓα+1αIa+gx−Γ2α+12αIa+gxx−aα.Now, using again the definition of stronglyα,m-convexity for f, the following inequality is valid:(43)∣f′t∣≤t−xb−xα∣f′b∣+m1−t−xb−xαf′xm−λmt−xb−xα1−t−xb−xαmb−xm2.On the same procedure as we did for (13) and (36), one can obtain following inequality from (14) and (43):(44)gFμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KbxEμ,β,lγ,δ,k,c,g;ϕ∣f′b∣gb−mf′xmgx−Γα+1b−xα∣f′b∣−mf′xmαIb−gx−λmb−x2mb−xαΓ2α+12αIb−gxb−xα−Γα+1αIb−gx.From (42) and (44), inequality (34) can be obtained.Corollary 15.
If we considerα=1 and λ=0 in (34), then the following inequality holds for the m-convex function:(45)gFμ,α′,l,a+ϕ,γ,δ,k,cf∗gx,ω;p+gFμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α′,lγ,δ,k,c,g;ϕgx−gaf′xm+∣f′a∣+KbxEμ,β,lγ,δ,k,c,g;ϕgb−gx∣f′b∣+f′xm.Remark 16.
(i) If we considerλ=0 in (34), ([28], Theorem 14]) is obtained
(ii) If we considerα,m=1,1 in (34), ([29], Theorem 6]) is obtained
(iii) If we considerϕt=tα′ and gx=x, in (34), then ([24], Theorem 5]) is obtained
(iv) If we considerλ=0, α,m=1,1 in (34), then ([11], Theorem 25]) is obtained
(v) If we considerλ=0, ϕt=Γα′tα′/k+1 and p=ω=0 in (34), then ([10], Theorem 11]) can be obtained
(vi) If we considerα′=β in the result of (v), then ([10], Corollary 12]) can be obtained
(vii) If we considerλ=0, ϕt=Γα′tα′+1 and α,m=1,1 in (34), then ([6], Theorem 11]) is obtained
(viii) If we considerα′=β in the result of (vii), then ([6], Corollary 12]) is obtained
(ix) If we considerλ=0, ϕt=Γα′tα′/k+1, α,m =(1,1), gx=x, and p=ω=0 in (34), then ([4], Theorem 11]) can be obtained
(x) If we considerα′=β in the result of (ix), then ([4], Corollary 4]) can be obtained
(xi) If we considerα′=β=k=1 and x=a+b/2, in the result of (ix), then ([4], Corollary 5]) can be obtained
(xii) If we considerλ=0, ϕt=Γα′tα′+1, gx=x, and p=ω=0 and α,m=1,1 in (34), then ([5], Theorem 11]) is obtained
(xiii) If we considerα′=β in the result of (xii), then ([5], Corollary 5]) can be obtained


## 3. Concluding Remarks

In this paper, integral inequalities for stronglyα,m-convex functions are given by utilizing unified integral operators. Results for m-convex functions are presented in particular in the form of corollaries. In the form of remarks, many published results are highlighted as consequences of presented results. The reader can get integral and fractional integral inequalities for convex functions, strongly convex functions, α,m-convex functions, m-convex functions, and strongly m-convex functions.



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