# Inequalities for Unified Integral Operators via Stronglyα,h‐m-Convexity

Authors: Zhongyi Zhang; Ghulam Farid; Kahkashan Mahreen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we give a generalized definition namely stronglyα,h‐m-convex function that unifies many known definitions. By applying this new definition, we present inequalities for unified integral operators which have connection with many of the well-known results for different kinds of convex functions. Moreover, this paper at once provides refinements and generalizations of a lot of fractional integral inequalities which are identified in remarks.


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## Body

## 1. Introduction

There are many applications of convexity in diverse fields of mathematics including operation research, mathematical statistics, optimization theory, and graph theory. In mathematical inequalities’ point of view, convex functions are very important. They are extended and generalized in different ways to obtain corresponding generalizations and extensions of well-known inequalities. For the detail study of different kinds of convex functions, we refer the readers to [1–7].In recent years, the researchers are working on fractional versions of mathematical inequalities by utilizing classical and new kinds fractional integral/derivative operators, see [8–11]. Also, several kinds of convex functions are applied to obtain these fractional versions, for example, see [1, 12–17] and references therein. The inequalities for fractional integrals and derivatives are very useful in the study of fractional differential equations. Using fractional differential and integral inequalities, qualitative properties of fractional differential equations involving the Riemann-Liouville and the Caputo derivatives can be found frequently in literature.Our objective is to investigate integral inequalities for newly defined function called stronglyα,h−m-convex function. Integral operators (5) and (6) are used to establish these inequalities, and they have interesting consequences for distinctive fractional inequalities for various types of functions. Next, we give some definitions of known generalized fractional integral operators which can be directly obtained from operators (5) and (6).Definition 1 (see [18]).
Letψ:a,b⟶ℝ be an integrable function and ξ be an increasing positive function defined on a,b has a continuous derivative ξ′ on a,b. The fractional integrals of a function ψ with respect to another function ξ on a,b of order μRμ>0 are defined by
(1)ξμIa+ψx=1Γμ∫axξx−ξtμ−1ξ′tψtdt,x>a,ξμIb−ψx=1Γμ∫xbξt−ξxμ−1ξ′tψtdt,x<b,here, Γ. represents the gamma function.One can see ak-analogue of Definition 1 in [16]. The following generalized integral operator is given in [19].Definition 2.
Letψ,ξ:a,b⟶ℝ,0<a<b be the functions such that ψ be positive and ψ∈L1a,b, and ξ be differentiable and increasing. Also let ϕ/x be an increasing function on a,∞. Then, for x∈a,b, the left and right integral operators are defined by
(2)Fa+ϕ,ξψx=∫axϕξx−ξtξx−ξtξ′tψtdt,x>a,Fb−ϕ,ξψx=∫xbϕξt−ξxξt−ξxξ′tψtdt,x<b.The Mittag-Leffler function was introduced in 1903, which is generalization of the exponential function just by a single parameter with a convergence condition. The further generalization by another parameter was given by Wiman; further, it was extended by Prabhakar and then by other authors; to see the importance of these extensions, we suggest the reader to [20, 21]. By utilizing an extended generalized Mittag-Leffler function, we have defined a fractional integral operator.Definition 3 (see [12]).
Letα,γ,μ,c,l,w∈ℂ, Rα,Rμ,Rl>0, Rc>Rγ>0 and p≥0, δ>0 with 0<k≤δ+Rμ. Let ψ∈L1a,b and x∈a,b. The generalized fractional integral operators εμ,α,l,w,a+γ,δ,k,cψ and εμ,α,l,w,b−γ,δ,k,cψ are defined by:
(3)εμ,α,l,w,a+γ,δ,k,cψx;p=∫axx−tα−1Eμ,α,lγ,δ,k,cwx−tμ;pψtdt,εμ,α,l,w,b−γ,δ,k,cψx;p=∫xbt−xα−1Eμ,α,lγ,δ,k,cwt−xμ;pψtdt,where
(4)Eμ,α,lγ,δ,k,ct;p=∑n=0∞ηpγ+nk,c−γηγ,c−γcnkΓμn+αtnlnδ,is the extended generalized Mittag-Leffler function.Unified integral operator is based on a kernel which also involves a real valued strictly increasing function along with two variables. This integral operator also unifies the above definitions.Definition 4 (see [22]).
Letψ,ξ:a,b⟶ℝ, 0<a<b be the functions such that ψ be positive and ψ∈L1a,b, and ξ be differentiable and strictly increasing. Also, let ϕ/x be differentiable and strictly increasing. Also, let a,∞ and τ,l,γ,c∈ℂ, Rτ,Rl>0, Rc>Rγ>0, p,μ,δ≥0 and 0<k≤δ+μ. Then, for x∈a,b, the left and right integral operators are defined by
(5)ξFμ,τ,l,a+ϕ,γ,δ,k,cψx,w;p=∫axKxyEμ,τ,lγ,δ,k,c,ξ;ϕψydξy,(6)ξFμ,τ,l,b−ϕ,γ,δ,k,cψx,w;p=∫xbKyxEμ,τ,lγ,δ,k,c,ξ;ϕψydξy,where we have
(7)KxyEμ,τ,lγ,δ,k,c,ξ;ϕ=ϕξx−ξyξx−ξyEμ,τ,lγ,δ,k,cwξx−ξyμ;p.Several recently defined fractional integrals studied in [12, 14, 18, 21, 23–30] can be reproduced from the above definition, see ([31], Remarks 6 and 7). The following results are obtained for strongly convex functions in [32].Theorem 5.
Letψ:a,b⟶ℝ be a positive, integrable, and strongly convex function with modulus λ≥0. Let ξ:a,b⟶ℝ be a strictly increasing and differentiable function, also let ϕ/x be an increasing function on a,b. If α,γ,η,c,l∈ℝ+, c>γ, p,μ,δ≥0 and 0<k≤δ+μ, then for x∈a,b, the following inequality holds:
(8)ξFμ,α,l,a+ϕ,γ,δ,k,cψx,w;p+ξFμ,η,l,b−ϕ,γ,δ,k,cψx,w;p≤KxaEμ,α,lγ,δ,k,c,ξ;ϕξx−ξaψx+ψa−λx−a2Ia,x,Idξ−a+xIa,x,ξ+KbxEμ,η,lγ,δ,k,c,ξ;ϕξb−ξxψb+ψx−λb−x2Ix,b,Idξ−x+bIx,b,ξ,where Id is the identity function and Ia,b,ψ≔∫abψtdt.Theorem 6.
Under the assumptions of Theorem5, in addition, if ψx=ψa+b−x, then, we have
(9)fa+b2ξFμ,α,l,b−ϕ,γ,δ,k,c1a,w;p+λ4ξFμ,α,l,b−ϕ,γ,δ,k,ca+b−2x2a,w;p+fa+b2ξFμ,η,l,a+ϕ,γ,δ,k,c1b,w;p+λ4ξFμ,η,l,a+ϕ,γ,δ,k,ca+b−2x2b,w;p≤ξFμ,α,l,b−ϕ,γ,δ,k,cψa,w;p+ξFμ,η,l,a+ϕ,γ,δ,k,cψb,w;p≤KbaEμ,α,lγ,δ,k,c,ξ;ϕ+KbaEμ,η,lγ,δ,k,c,ξ;ϕξb−ξaψb+ψa−b−aλ2Ia,b,Idξ−a+bIa,b,ξ.Theorem 7.
Letψ:a,b⟶ℝ be a differentiable function. If ∣ψ′∣ is strongly convex with modulus λ≥0 and ξ:a,b⟶ℝ be strictly increasing and differentiable, also let ϕ/x be an increasing function on a,b. If α,γ,η,c,l∈ℝ+, c>γ, p,μ,δ≥0 and 0<k≤δ+μ, then for x∈a,b, the following inequality holds:
(10)ξFμ,α,l,a+ϕ,γ,δ,k,cψ∗ξx,w;p+ξFμ,η,l,b−ϕ,γ,δ,k,cψ∗ξx,w;p≤KxaEμ,α,lγ,δ,k,c,ξ;ϕ×ξx−ξaψ′x+ψ′a−λx−a2Ia,x,Idξ−a+xIa,x,ξ+KbxEμ,η,lγ,δ,k,c,ξ;ϕξb−ξxψ′b+ψ′x−λb−x2Ix,b,Idξ−x+bIx,b,ξ.Next, we give some definitions of convex functions. The definition ofh‐m-convex function is given as follows:Definition 8 (see [5]).
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a nonnegative function. A nonnegative function ψ:0,b⟶ℝ is called h‐m-convex function if
(11)ψζx+m1−ζy≤hζψx+mh1−ζψy,holds for all x,y∈0,b, m∈0,1, and ζ∈0,1.Remark 9.
(i)
Form=1, (11) gives the definition of h-convex function(ii)
Forhζ=ζ, (11) gives the definition of m-convex function(iii)
Forhζ=ζ and m=1, (11) gives the definition of convex function(iv)
Forhζ=1 and m=1, (11) gives the definition of p-function(v)
Forhζ=ζs and m=1, (11) gives the definition of s-convex function(vi)
Forhζ=1/ζ and m=1, (11) gives the definition of Godunova-Levin function(vii)
Forhζ=1/ζs and m=1, (11) gives the definition of s-Godunova-Levin function of secondDefinition 10 (see [4]).
A functionψ:0,b⟶ℝ,b>0 is said to be α,m-convex if
(12)ψζx+m1−ζy≤ζαψx+m1−ζαψy,holds for all x,y∈0,bζ∈0,1 and α,m∈0,12.Remark 11.
(i)
Forα,m=1,m, (12) provides m-convex function(ii)
Forα,m=1,1, (12) provides convex function(iii)
Forα,m=1,0, (12) provides star-shaped functionDefinition 12 see ([33]).
A functionψ:0,b⟶ℝ,b>0 is said to be s,m-convex, where s,m∈0,12 if
(13)ψζx+m1−ζy≤ζsψx+m1−ζsψy,holds for all x,y∈0,b and ζ∈0,1.The following definition unifiesh‐m-convex, s,m-convex, and α,m-convex functions in a single inequality.Definition 13.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a nonnegative function. A nonnegative function ψ:0,b⟶ℝ is said to be α,h‐m-convex function if
(14)ψζx+m1−ζy≤hζαψx+mh1−ζαψy,holds for all x,y∈0,b,ζ∈0,1, α,m∈0,12.Next, we give definitions of strongly convex, stronglys,m-convex, and strongly α,m-convex functions.Definition 14 (see [34]).
LetI be a nonempty convex subset of normed space X,.. A real valued function ψ is said to be strongly convex, with modulus λ≥0, on I if for each a,b∈I and ζ∈0,1,
(15)ψζx+1−ζy≤ζψx+1−ζψy−λζ1−ζb−a2.Definition 15 (see [35]).
A functionψ:0,+∞⟶ℝ is said to be strongly s,m-convex function, with modulus λ≥0, for s,m∈0,12, if
(16)ψζx+m1−ζy≤ζsψx+m1−ζsψy−λmζ1−ζy−x2,holds for all x,y∈0,+∞ and ζ∈0,1.Definition 16 (see [36]).
A functionψ:0,b⟶ℝ,b>0 is said to be strongly α,m-convex, where α,m∈0,12 if
(17)ψζx+m1−ζy≤ζαψx+m1−ζαψy−λmζα1−ζαy−x2,holds for all x,y∈0,b and ζ∈0,1.Next, we give a property of the kernel given in (7), which will be useful for finding the results of this paper.P: Let ξ and ϕ/I be increasing functions. Then, for u<t<v, u,v∈a,b, the kernel KuvEμ,τ,lγ,δ,k,c,ξ;ϕ satisfies the following inequality:
(18)KtuEμ,τ,lγ,δ,k,c,ξ;ϕξ′t≤KvuEμ,τ,lγ,δ,k,c,ξ;ϕξ′t.It is easy to prove by using the following inequalities:(19)ϕξt−ξuξt−ξuξ′t≤ϕξv−ξuξv−ξuξ′t,Eμ,τ,lγ,δ,k,cwξt−ξuμ;p≤Eμ,τ,lγ,δ,k,cwξv−ξuμ;p.Ifξ and ϕ/I are of opposite monotonicities, then (18) holds in reverse direction. For further properties, see [37].In Section2, we will define a new notion of strongly α,h‐m-convex function which unifies several kinds of convex functions in a single inequality. By applying this new definition, we give generalizations of results for strongly convex functions. The results established here will produce generalizations and refinements of fractional integral inequalities for different kinds of convex and strongly convex functions which have been published in various papers.


## 2. Main Results

We give the definition of a function will be called stronglyα,h‐m-convex function.Definition 17.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a nonnegative function. A nonnegative function ψ:0,b⟶ℝ is said to be strongly α,h‐m-convex function with modulus λ≥0 if
(20)ψζx+m1−ζy≤hζαψx+mh1−ζαψy−mλhζαh1−ζαy−x2,holds for all x,y∈0,b,ζ∈0,1, α,m∈0,12.The definition of stronglyh‐m-convexity can be achieved by taking α=1 in (20).Definition 18.
LetJ⊆ℝ be an interval containing 0,1 and let h:J⟶ℝ be a nonnegative function. A nonnegative function ψ:0,b⟶ℝ is said to be strongly h‐m-convex function with modulus λ≥0 if
(21)ψζx+m1−ζy≤hζψx+mh1−ζψy−mλhζh1−ζy−x2,holds for all x,y∈0,b,ζ∈0,1, m∈0,1.One can obtain from (20) definitions of strongly convex, strongly s-convex, strongly m-convex, strongly h-convex, strongly α,m-convex, and strongly s,m-convex functions.Theorem 19.
Letψ∈L1a,b be a positive strongly α,h‐m-convex function with modulus λ≥0, m∈0,1,0<a<mb. Let ξ be strictly increasing and differentiable function, also let ϕ/x be an increasing function on a,b and hxhy≤hx+y. If τ,η,l,γ,c∈ℝ+, c>γ, p,μ,δ≥0 and 0<k≤δ+μ, then for x∈a,b, the following inequality holds:
(22)ξFμ,τ,l,a+ϕ,γ,δ,k,cψx,w;p+ξFμ,η,l,b−ϕ,γ,δ,k,cψx,w;p≤KxaEμ,τ,lγ,δ,k,c,ξ;ϕx−aψaXxarα,h;ξ′+mψxmXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a+KbxEμ,η,lγ,δ,k,c,ξ;ϕb−x×ψbXxbrα,h;ξ′+mψxmXxb1−rα,h;ξ′−λbm−x2h1ξb−ξxmb−x,while Xxarα,h;ξ′=∫01hrαξ′x−rx−adr, Xxa1−rα,h;ξ′=∫01h1−rαξ′x−rx−adr.Proof.
UsingP, we can write the following inequalities
(23)KxtEμ,τ,lγ,δ,k,c,ξ;ϕξ′t≤KxaEμ,τ,lγ,δ,k,c,ξ;ϕξ′t,t∈a,x,(24)KtxEμ,τ,lγ,δ,k,c,ξ;ϕξ′t≤KbxEμ,η,lγ,δ,k,c,ξ;ϕξ′t,t∈x,b.
Using stronglyα,h‐m-convexity of ψ, we have
(25)ψt≤hx−tx−aαψa+mh1−x−tx−aαψxm−λx−am2mhx−tx−aαh1−x−tx−aα,(26)ψt≤ht−xb−xαψb+mh1−t−xb−xαψxm−λbm−x2mht−xb−xαh1−t−xb−xα.
From (23) and (25), the following inequality is obtained:
(27)∫axKxtEμ,τ,lγ,δ,k,c,ξ;ϕψtdξt≤ψaKxaEμ,τ,lγ,δ,k,c,ξ;ϕ×∫axhx−tx−aαdξt+mψxmKxaEμ,τ,lγ,δ,k,c,ξ;ϕ×∫axh1−x−tx−aαdξt−λbm−x2mKxaEμ,τ,lγ,δ,k,c,ξ;ϕ×∫axh1−x−tx−aαhx−tx−aαdξt.
By settingr=x−t/x−a on the right side and using (5) on left side of above inequality, we get
(28)ξFμ,τ,l,a+ϕ,γ,δ,k,cψx,w;p≤KxaEμ,τ,lγ,δ,k,c,ξ;ϕx−a×ψa∫01hrαξ′x−rx−adr+mψxm∫01h1−rαξ′x−rx−adr−λx−am2m∫01h1−rαhrαξ′x−rx−adr.
The inequality (28) can take the following form:
(29)ξFμ,τ,l,a+ϕ,γ,δ,k,cψx,w;p≤KxaEμ,τ,lγ,δ,k,c,ξ;ϕx−a×ψaXxarα,h;ξ′+mψxmXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a.
On the other hand, multiplying (24) and (26), and adopting the same pattern as we did for (23) and (25), the following inequality holds true:
(30)ξFμ,η,l,b−ϕ,γ,δ,k,cψx,w;p≤KbxEμ,η,lγ,δ,k,c,ξ;ϕb−x×ψb∫01hrαξ′x−rx−bdr+mψxm∫01h1−rαξ′x−rx−bdr−λbm−x2m∫01h1−rαhrαξ′x−rx−bdr.
The inequality (30) can take the following form:
(31)ξFμ,η,l,b−ϕ,γ,δ,k,cψx,w;p≤KbxEμ,η,lγ,δ,k,c,ξ;ϕb−x×ψbXxbrα,h;ξ′+mψxmXxb1−rα,h;ξ′−λbm−x2h1ξb−ξxmb−x.
By adding (29) and (31), (22) can be achieved.☐Corollary 20.
Forw=p=0, (22) gives the following inequality obtained for fractional integral operator defined in [19]:
(32)Fa+ϕ,ξψxΓτ+Fb−ϕ,ξψxΓη≤ϕξx−ξax−aΓτξx−ξaψaXxarα,h;ξ′+mψxmXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a+ϕξb−ξxb−xΓηξb−ξx×ψbXxbrα,h;ξ′+mψxmXxb1−rα,h;ξ′−λbm−x2h1ξb−ξxmb−x.Remark 21.
(i)
Forλ=0, (22) gives ([38], Theorem 1)(ii)
Forτ=η and hζ=ζ, (22) gives ([39], Theorem 1)(iii)
Forλ=0, τ=η, α=m=1, and hζ=ζ, (22) gives ([31], Theorem 8)(iv)
Forλ=0, ϕζ=Γτζτ/k/kΓkτ, hζ=ξt=ζ, α=m=1, and w=p=0, (22) gives ([40], Theorem 1)(v)
Forλ=0, τ=η, the result of (iv) gives ([40], Corollary 1)(vi)
Forλ=0, k=1, and x=a or x=b, the result of (v) gives ([40], Corollary 2)(vii)
Forλ=0, k=1, and x=a+b/2, the result of (v) gives ([40], Corollary 3)(viii)
Forλ=0, ϕζ=ζτ, α=1, and ξt=ζ, (22) gives ([41], Theorem 1)(ix)
Forλ=0, ϕζ=ζτ, α=m=1, and hζ=ξt=ζ, (22) gives ([41], Corollary 1)(x)
Forλ=0, ϕζ=Γτζτ, α=1, w=p=0, and ξζ=ζ, (22) gives ([42], Theorem 2.1)(xi)
Forλ=0, τ=η, the result of (x) gives ([42], Corollary 2.2)(xii)
Forλ=0, τ=η, ϕζ=Γτζτ, w=p=0, ξζ=ζ, and α=1 and using ([12], Remark 11), (22) gives ([42], Corollary 2.3)(xiii)
Forλ=0, ϕζ=Γτζτ, α=1, w=p=0, ξζ=ζ, and hζ=1, (22) gives inequality (26) of ([42], Corollary 2.4) similarly, under the same assumptions along with hζ=ζp, (22) gives inequality (27) of ([42], Corollary 2.4)(xiv)
Forλ=0, ϕζ=Γτζτ, w=p=0, α=m=1, and ξt=hζ=ζ, (22) gives ([43], Theorem 1)(xv)
Forλ=0, τ=η, the result of (xiv) gives ([43], Corollary 1)(xvi)
Forλ=0, ϕζ=Γτζτ, w=p=0, α=m=1, and hζ=ζ, (22) gives ([44], Theorem 1)(xvii)
Forλ=0, τ=η, ϕζ=Γτζτ, w=p=0, α=m=1, and hζ=ζ, (22) gives ([44], Corollary 1)(xviii)
Forλ=0, ϕζ=ζτ, ξζ=ζ, and hζ=ζs, α=m=1, (22) gives ([45], Theorem 2.1)(xix)
Forλ=0, τ=η, ϕζ=ζτ, ξζ=ζ, α=m=1, and hζ=ζs, (22) gives ([45], Corollary 2.1)(xx)
Forλ=0, w=p=0, α=1, and hζ=ζs, (22) gives ([46], Theorem 1)(xxi)
Forλ=0, ϕζ=ζτ, hζ=ζs, α=1, and ξζ=ζ, (22) gives ([47], Theorem 1)(xxii)
Forλ=0, τ=η, the result of (xxi) gives ([47], Corollary 1)(xxiii)
Forλ=0, ϕζ=Γτζτ/k/kΓkτ, hζ=z, and w=p=0, (22) gives ([15], Theorem 1)(xxiv)
Forλ=0, τ=η, the result of (xxiii) gives ([15], Corollary 1)(xxv)
Forϕζ=ζτ, ξζ=ζ, and hζ=ζ, (22) gives ([36], Theorem 4)(xxvi)
Forτ=η, the result of (xxiv) gives ([36], Corollary 1)(xxvii)
Forψ∈L∞a,b, the result of (xxiv) gives ([36], Corollary 2)(xxviii)
Forτ=η, the result of (xxvii) gives ([36], Corollary 3)(xxix)
Forϕζ=ζτ, ξζ=ζ, and λ=0, (22) gives ([48], Theorem 1)(xxx)
Forτ=η, the result of (xxix) gives ([48], Corollary 1)(xxxi)
Forψ∈L∞a,b, the result of (xxix) gives ([48], Corollary 2)(xxxii)
Forτ=η, the result of (xxxi), gives ([48], Corollary 3)For the proof of next theorem, we need the following lemma.Lemma 22.
Letψ:a,b⟶ℝ, be a strongly α,h‐m-convex function with modulus λ≥0, m∈0,1,0≤a<mb. If ψx=ψa+mb−x/m, then the following inequality holds:
(33)ψa+mb2≤h12α+mh2α−12αψx−λmh12αh2α−12αa−x+mb−mx2.Proof.
Asψ is strongly α,h‐m-convex function, we have
(34)ψa+mb2≤h12αψ1−ta+mtb+mh2α−12αψt+m1−tbm−λmh12αh2α−12αt1+ma−mb+mb−ma2.
Letx=a1−t+mtb. Then, we have a+mb−x=ta+m1−tb, and using ψa+mb−x/m=ψx, the inequality (33) is obtained.☐The upcoming theorem provides the Hadamard inequality for stronglyα,h‐m-convex function.Theorem 23.
Under the assumptions of Theorem19, in addition, if ψx=ψa+mb−x/m, then, we have
(35)1h1/2α+mh2α−1/2αψa+mb2ξFμ,τ,l,b−ϕ,γ,δ,k,c1a,w;p+ξFμ,η,l,a+ϕ,γ,δ,k,c1b,w;p+λmh12αh2α−12αξFμ,τ,l,b−ϕ,γ,δ,k,ca−x+mb−mx2a,w;p+ξFμ,η,l,a+ϕ,γ,δ,k,ca−x+mb−mx2b,w;p≤ξFμ,η,l,a+ϕ,γ,δ,k,cψb,w;p+ξFμ,τ,l,b−ϕ,γ,δ,k,cψa,w;p≤b−aKbaEμ,τ,lγ,δ,k,c,ξ;ϕ+KbaEμ,η,lγ,δ,k,c,ξ;ϕψbXbarα,h;ξ′+mψamXba1−rα,h;ξ′−λb−ma2h1ξb−ξamb−a.Proof.
UsingP, we can write the following inequalities:
(36)KxaEμ,τ,lγ,δ,k,c,ξ;ϕξ′x≤KbaEμ,τ,lγ,δ,k,c,ξ;ϕξ′x,x∈a,b,(37)KbxEμ,η,lγ,δ,k,c,ξ;ϕξ′x≤KbaEμ,η,lγ,δ,k,c,ξ;ϕξ′x,x∈a,b.
Using stronglyα,h‐m-convexity of ψ, we have
(38)ψx≤hx−ab−aαψb+mh1−x−ab−aαψam−λa−bm2mhx−ab−aαh1−x−ab−aα.
Multiplying (36) and (38) and integrating the resulting inequality over a,b, we obtain
(39)∫abKxaEμ,τ,lγ,δ,k,c,ξ;ϕψxdξx≤ψbKbaEμ,τ,lγ,δ,k,c,ξ;ϕ×∫abhx−ab−aαdξx+mψamKbaEμ,τ,lγ,δ,k,c,ξ;ϕ∫abh1−x−ab−aαdξx−λa−bm2m∫abhx−ab−aαh1−x−ab−aαdξx.
By settingr=x−a/b−a on the right side and using (5) on left side of above inequality, we get
(40)ξFμ,τ,l,b−ϕ,γ,δ,k,cψa,w;p≤KbaEμ,τ,lγ,δ,k,c,ξ;ϕb−a×ψb∫01hrαξ′a+rb−adr+mψam∫01h1−rαξ′a+rb−adr−λa−bm2m∫01hrαh1−rαξ′a+rb−adr.
The inequality (40) can take the following form:
(41)ξFμ,τ,l,b−ϕ,γ,δ,k,cψa,w;p≤KbaEμ,τ,lγ,δ,k,c,ξ;ϕb−a×ψbXbarα,h;ξ′+mψamXba1−rα,h;ξ′−λa−bm2h1ξb−ξamb−a.
Adopting the same pattern of simplification as we did for (36) and (38), the following inequality can be observed for (38) and (37):
(42)ξFμ,η,l,a+ϕ,γ,δ,k,cψb,w;p≤KbaEμ,η,lγ,δ,k,c,ξ;ϕb−a×ψbXbarα,h;ξ′+mψamXba1−rα,h;ξ′−λa−bm2h1ξb−ξamb−a.
By adding (41) and (42), following inequality can be achieved:
(43)ξFμ,η,l,a+ϕ,γ,δ,k,cψb,w;p+ξFμ,τ,l,b−ϕ,γ,δ,k,cψa,w;p≤b−aKbaEμ,τ,lγ,δ,k,c,ξ;ϕ+KbaEμ,η,lγ,δ,k,c,ξ;ϕψbXbarα,h;ξ′+mψamXba1−rα,h;ξ′−λa−bm2h1ξb−ξamb−a.
Multiplying both sides of (33) by KxaEμ,τ,lγ,δ,k,c,ξ;ϕdξx and integrating over a,b, one can get
(44)ψa+mb2∫abKxaEμ,τ,lγ,δ,k,c,ξ;ϕdξx≤h12α+mh2α−12α∫abKxaEμ,τ,lγ,δ,k,c,ξ;ϕψxdξx−λmh12αh2α−12α∫abKxaEμ,τ,lγ,δ,k,c,ξ;ϕa−x+mb−mx2dξx.
By using the Definition4, one can obtain the following inequality:
(45)1h1/2α+mh2α−1/2αψa+mb2ξFμ,τ,l,b−ϕ,γ,δ,k,c1a,w;p+λmh12α×h2α−12αξFμ,τ,l,b−ϕ,γ,δ,k,ca−x+mb−mx2a,w;p≤ξFμ,τ,l,b−ϕ,γ,δ,k,cψa,w;p.
Now, multiplying byKbxEμ,ηγ,δ,k,c,ξ;ϕdξx on both sides of (33), then integrating over a,b, we get
(46)1h1/2α+mh2α−1/2αψa+mb2ξFμ,η,l,a+ϕ,γ,δ,k,c1b,w;p+λmh12α×h2α−12αξFμ,η,l,a+ϕ,γ,δ,k,ca−x+mb−mx2b,w;p≤ξFμ,η,l,a+ϕ,γ,δ,k,cψb,w;p.
From (43), (45), and (46), inequality (35) can be achieved.☐Corollary 24.
Forw=p=0, (35) gives the following inequality obtained for the fractional integral operator that defined in [19]:
(47)1h1/2α+mh2α−1/2αψa+mb2Fb−ϕ,ξ1aΓτ+Fa+ϕ,ξ1bΓη+λmh12αh2α−12αFb−ϕ,ξa−x+mb−mx2aΓτ+Fa+ϕ,ξa−x+mb−mx2bΓη≤Fa+ϕ,ξψbΓη+Fb−ϕ,ξψaΓτ≤b−aξb−ξaξb−ξa1Γτ+1Γη×ψbXbarα,h;ξ′+mψamXba1−rα,h;ξ′−λb−ma2h1ξb−ξamb−a.Remark 25.
(i)
Forλ=0, τ=η, α=m=1, and hζ=ζ, (35) gives ([31], Theorem 22)(ii)
Forλ=0, ϕζ=Γτζτ/k+1, hζ=ξζ=ζ, α=m=1, and w=p=0, (35) gives ([40], Theorem 3)(iii)
Forλ=0, τ=η, the result of (ii) gives ([40], Corollary 6)(iv)
Forλ=0, ϕζ=Γτζτ+1, w=p=0, α=m=1, and ξζ=hζ=ζ, (35) gives ([43], Theorem 3)(v)
Forλ=0, τ=η, and ϕζ=Γτζτ in the result of (v) gives ([43], Corollary 6)(vi)
Forλ=0, ϕζ=Γτζτ+1, w=p=0, α=m=1, and hζ=ζ, (35) gives ([44], Theorem 3)(vii)
Forλ=0, τ=η, the result of (vii) gives ([44], Corollary 3)(viii)
Forλ=0, ϕζ=ζτ+1, ξζ=ζ, and hζ=ζs, α=m=1, (35) gives ([45], Theorem 2.4)(ix)
Forλ=0, τ=η, the result of (ix) gives ([45], Corollary 2.6)(x)
Forϕζ=ζτ+1, ξζ=ζ, and hζ=ζ, (35) gives ([36], Theorem 6)(xi)
Forτ=η, the result of (xi) gives ([36], Corollary 5)(xii)
Forϕζ=ζτ+1, ξζ=ζ, and λ=0, (35) gives ([48], Theorem 4)(xiii)
Forτ=η, the result of (xiii) gives ([48], Corollary 5)Theorem 26.
Letψ:a,b⟶ℝ be a differentiable function such that ∣ψ′∣ is strongly α,h‐m-convex with modulus λ≥0, m∈0,1,0<a<mb. Let ξ:a,b⟶ℝ be strictly increasing and differentiable function, also let ϕ/x be a function which is increasing on the interval a,b and hxhy≤hx+y. If c>γ, p,μ,δ≥0, τ,η,l,γ,c∈ℝ+, and 0<k≤δ+μ, then for x∈a,b the following inequality holds:
(48)ξFμ,τ,l,a+ϕ,γ,δ,k,cψ∗ξx,w;p+ξFμ,η,l,b−ϕ,γ,δ,k,cψ∗ξx,w;p≤KxaEμ,τ,lγ,δ,k,c,ξ;ϕ×x−aψ′aXxarα,h;ξ′+mψ′xmXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a+KbxEμ,η,lγ,δ,k,c,ξ;ϕb−x×ψ′bXxbrα,h;ξ′+mψ′xmXxb1−rα,h;ξ′−λb−mb2h1ξb−ξxmb−x,where
(49)ξFμ,τ,l,a+ϕ,γ,δ,k,cψ∗ξx,w;p≔∫axKxtEμ,τ,lγ,δ,k,c,ξ;ϕψ′tdξt,ξFμ,η,l,b−ϕ,γ,δ,k,cψ∗ξx,w;p≔∫xbKtxEμ,τ,lγ,δ,k,c,ξ;ϕψ′tdξt.Proof.
Since∣ψ′∣ is strongly α,h‐m-convex function, one can have
(50)∣ψ′t∣≤hx−tx−aα∣ψ′a∣+mh1−x−tx−aαψ′xm−λx−am2mhx−tx−aαh1−x−tx−aα.
The inequality (50) can take the following form:
(51)−hx−tx−aαψ′a+mh1−x−tx−aαψ′xm−λx−am2mhx−tx−aαh1−x−tx−aα≤ψ′t≤hx−tx−aαψ′a+mh1−x−tx−aαψ′xm−λx−am2mhx−tx−aαh1−x−tx−aα.
From inequality (51), we have
(52)ψ′t≤hx−tx−aα∣ψ′a∣+mh1−x−tx−aαψ′xm−λx−am2mhx−tx−aαh1−x−tx−aα.
Multiplying (23) and (52) and integrating over a,x, we obtain
(53)∫axKxtEμ,τ,lγ,δ,k,c,ξ;ϕψ′tdξt≤∣ψ′a∣KxaEμ,τ,lγ,δ,k,c,ξ;ϕ×∫axhx−tx−aαdξt+mψ′xmKxaEμ,τ,lγ,δ,k,c,ξ;ϕ×∫axh1−x−tx−aαdξt−λbm−x2mKxaEμ,τ,lγ,δ,k,c,ξ;ϕ×∫axh1−x−tx−aαhx−tx−aαdξt,
which gives(54)ξFμ,τ,l,a+ϕ,γ,δ,k,cψ∗ξx,w;p≤KxaEμ,τ,lγ,δ,k,c,ξ;ϕx−a×ψ′aXxarα,h;ξ′+mψ′xmXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a.
Using the other inequality of (51) and doing on the same way as adopted for the right hand inequality, one can get
(55)ξFμ,τ,l,a+ϕ,γ,δ,k,cψ∗ξx,w;p≥−KxaEμ,τ,lγ,δ,k,c,ξ;ϕx−a×ψ′aXxarα,h;ξ′+mψ′xmXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a.
From (54) and (55), the following inequality is observed:
(56)ξFμ,τ,l,a+ϕ,γ,δ,k,cψ∗ξx,w;p≤KxaEμ,τ,lγ,δ,k,c,ξ;ϕx−a×ψ′aXxarα,h;ξ′+mψ′xmXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a.
By applying stronglyα,h‐m-convexity of ∣ψ′∣, one can get
(57)∣ψ′t∣≤ht−xb−xα∣ψ′b∣+mh1−t−xb−xαψ′xm−λbm−x2mht−xb−xαh1−t−xb−xα.
By following the same steps as we did for (23) and (50), from (24) and (57), one can get the following inequality:
(58)ξFμ,η,l,b−ϕ,γ,δ,k,cψ∗ξx,w;p≤KbxEμ,η,lγ,δ,k,c,ξ;ϕb−x×ψ′bXxbrα,h;ξ′+mψ′xmXxb1−rα,h;ξ′−λbm−x2h1ξb−ξxmb−x.
By adding (56) and (58), inequality (48) can be achieved.☐Corollary 27.
Forw=p=0, (48) gives the following inequality obtained for the fractional integral operator that defined in [19]:
(59)Fa+ϕ,ξψ∗ξxΓτ+Fb−ϕ,ξψ∗ξxΓη≤ϕξx−ξax−aΓτξx−ξa×ψ′aXxarα,h;ξ′+mψ′x/mXxa1−rα,h;ξ′−λx−ma2h1ξx−ξamx−a+ϕξb−ξxb−xΓηξb−ξx×ψ′bXxbrα,h;ξ′+mψ′x/mXxb1−rα,h;ξ′−λb−mb2h1ξb−ξxmb−x.Remark 28.
(i)
Forλ=0, (48) gives ([38], Theorem 4)(ii)
Forλ=0, τ=η, and hζ=ζ, ((48) gives ([39], Theorem 3)(iii)
Forλ=0, τ=η, α=m=1, and hζ=ζ, (48) gives ([31], Theorem 25)(iv)
Forλ=0, ϕζ=Γτζτ/k+1, hζ=ξζ=ζ, α=m=1, and w=p=0, (48) gives ([40], Theorem 2)(v)
Forλ=0, τ=η, the result of (iv) gives ([40], Corollary 4)(vi)
Forλ=0, ϕζ=ζτ, α=1, and ξζ=ζ, (48) gives ([41], Theorem 2)(vii)
Forλ=0, m=1, and hζ=ζ, the result of (vi) gives ([41], Corollary 2)(viii)
Forλ=0, ϕζ=Γτζτ+1, α=1, w=p=0, and ξζ=ζ, (48) gives ([42], Theorem 2.6)(ix)
Forλ=0, τ=η, the result of (viii) gives ([42], Corollary 2.7)(x)
Forλ=0, ϕζ=Γτζτ+1, w=p=0, α=m=1, and ξζ=hζ=ζ, (48) gives ([43], Theorem 2)(xi)
Forλ=0, τ=η, and ϕζ=Γτζτ, the result of (x) gives ([43], Corollary 4)(xii)
Forλ=0, w=p=0, ϕζ=Γτζτ+1, α=m=1, and hζ=ζ, (48) gives ([44], Theorem 2)(xiii)
Forλ=0, τ=η, the result of (xii) gives ([44], Corollary 2).(xiv)
Forλ=0, ξζ=ζ and hζ=ζs, ϕζ=ζτ+1, α=m=1, (48) gives ([45], Theorem 2.3)(xv)
Forλ=0, τ=η, the result of (xiv) gives ([45], Corollary 2.5)(xvi)
Forλ=0, w=p=0, α=1 and hζ=ζs, (48) gives ([46], Theorem 2)(xvii)
Forλ=0, hζ=ζs, ϕζ=Γτζτ, α=1, and ξζ=ζ, (48) gives ([47], Theorem 3)(xviii)
Forλ=0, τ=η, the result of (xvii) gives ([47], Corollary 5)(xix)
Forλ=0, hζ=ζ, ϕζ=Γτζτ/k+1, and w=p=0, (48) gives ([15], Theorem 2)(xx)
Forλ=0, τ=η, the result of (xix) gives ([15], Corollary 2)(xxi)
Forϕζ=ζτ, ξζ=ζ, and hζ=ζ, (48) gives ([36], Theorem 5)(xxii)
Forτ=η in the result of (xxi) gives ([36], Corollary 4)(xxiii)
Forϕζ=ζτ+1, ξζ=ζ, and λ=0, (48) gives ([48], Theorem 3)(xxiv)
Forτ=η, the result of (xxiii) gives ([48], Corollary 4)


## 3. Concluding Remarks

A new definition is given and utilized to obtain some integral inequalities via a unified integral operator. The established results provide generalizations of many well-known inequalities. They also give refinements of recently published results for convex,m-convex, h-convex, s-convex, α,m-convex, s,m-convex, h‐m-convex and α,h‐m-convex functions. The reader also can obtain more fractional integral inequalities by setting appropriate functions and parameters involved in the kernel of unified integral operators.



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*Source: 6675826-2021-06-04.xml*