# Refinements of Some Integral Inequalities fors,m-Convex Functions

Authors: Ghulam Farid; Yu-Ming Chu; Maja Andrić; Chahn Yong Jung; Josip Pečarić; Shin Min Kang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the refinements of integral inequalities for all those types of convex functions are given which can be obtained froms,m-convex functions. These inequalities not only provide refinements of bounds for unified integral operators but also for various associated fractional integral operators containing Mittag–Leffler function. At the same time, presented results give generalizations of many known fractional integral inequalities.


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## Body

## 1. Introduction

The following fractional integral operator is the well-known Riemann–Liouville fractional integral operator.Definition 1.
(see [1]). Let f∈L1a,b. Then, Riemann–Liouville fractional integrals of order μ where ℜμ>0 are defined as follows:(1)Iμa+fx=1Γμ∫axx−tμ−1ftdt,x>a,Iμb−fx=1Γμ∫xbt−xμ−1ftdt,x<b,where Γ⋅ is the gamma function.
Next, generalizations of Riemann–Liouville fractional integral operators are given.Definition 2.
(see [2]). Let f: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 the 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(2)Igμa+fx=1Γμ∫axgx−gtμ−1g′tftdt,x>a,(3)Igμb−fx=1Γμ∫xbgt−gxμ−1g′tftdt,x<b,where Γ⋅ is the gamma function.
Ak-analogue of the above definition is given as follows.Definition 3.
(see [3]). Let f: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,k>0 are defined by(4)Igμb−kfx=1kΓkμ∫axgx−gtμ/k−1g′tftdt,x>a,Igμb−kfx=1kΓkμ∫xbgt−gxμ/k−1g′tftdt,x<b,where Γkx=∫0∞tx−1e−tk/kdt,ℜx>0.
The following integral operator is given in [4].Definition 4.
Letf,g:a,b⟶ℝ,0<a<b, be the functions such that f be positive and f∈L1a,b and g be differentiable and strictly 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(5)Fa+ϕ,gfx=∫axKgx,t;ϕftdgt,x>a,Fb−ϕ,gfx=∫xbKgt,x;ϕftdgt,x<b,where Kgx,y;ϕ=ϕgx−gy/gx−gy.
A fractional integral operator containing an extended generalized Mittag–Leffler function in its kernel is defined as follows.Definition 5.
(see [5]). Let ω,μ,α,l,γ,c∈ℂ, ℜμ,ℜα,ℜl>0, and 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(6)ϵμ,α,l,ω,a+γ,δ,k,cfx;p=∫axx−tα−1Eμ,α,lγ,δ,k,cωx−tμ;pftdt,(7)ϵμ,α,l,ω,b−γ,δ,k,cfx;p=∫xbt−xα−1Eμ,α,lγ,δ,k,cωt−xμ;pftdt,where(8)Eμ,α,lγ,δ,k,ct;p=∑n=0∞βpγ+nk,c−γβγ,c−γcnkΓμn+αtnlnδ,is the extended generalized Mittag–Leffler function. For further study of the Mittag–Leffler function, see [6, 7]. cnk is the Pochhammer symbol defined by cnk=Γc+nk/Γc, and βp is the extended beta function given by(9)βpx,y=∫01tx−11−ty−1e−p/t1−tdt,x,y,p∈ℝ+.
The following identities for the constant function are obtained in [8] (see also [9]):(10)Jα,a+x;p≔ϵμ,α,l,ω,a+γ,δ,k,c1x;p=x−aαEμ,α+1,lγ,δ,k,cwx−aμ;p,Jβ,b−x;p≔ϵμ,β,l,ω,b−γ,δ,k,c1x;p=b−xβEμ,β+1,lγ,δ,k,cwb−xμ;p.
Recently, a unified integral operator is defined as follows.Definition 6.
(see [10]). Let f,g:a,b⟶ℝ, 0<a<b, be the functions such that f be positive and f∈L1a,b and g be differentiable and strictly increasing. Also, let ϕ/x be an increasing function on a,∞ and α,l,γ,c∈ℂ, p,μ,δ≥0, and 0<k≤δ+μ. Then, for x∈a,b, the left and the right integral operators are defined by(11)Fgμ,α,l,a+ϕ,γ,δ,k,cfx,ω;p=∫axKxyEμ,α,lγ,δ,k,c,g;ϕfydgy,(12)Fgμ,α,l,b−ϕ,γ,δ,k,cfx,ω;p=∫xbKyxEμ,β,lγ,δ,k,c,g;ϕfydgy,where the involved kernel is defined by(13)KxyEμ,α,lγ,δ,k,c,g;ϕ=ϕgx−gygx−gyEμ,α,lγ,δ,k,cωgx−gyμ;p.
The known fractional integrals studied in [2, 11–22] can be reproduced from the above definition, see [23], Remarks 6 and 7.
The aim of this study is to obtain the bounds of all known fractional integral operators defined in [2, 11–22] in a unified form for strongly s,m-convex functions. In the result, we get refinements of many known integral and fractional integral inequalities. Next, we recall definitions of convex, strongly convex, s-convex, m-convex, s,m-convex, and strongly s,m-convex functions.Definition 7.
(see [24]). A function f:I⟶ℝ is said to be a convex function if the inequality(14)fta+1−tb≤tfa+1−tfb,holds for all a,b∈I and t∈0,1.
The concept of a strongly convex function is defined as follows.Definition 8.
(see [25]). Let I be a nonempty convex subset of a normed space. A real-valued function f is said to be strongly convex with modulus λ≥0 on I if for each a,b∈I and t∈0,1, we have(15)fta+1−tb≤tfa+1−tfb−λt1−tb−a2.
A generalization of the convex function defined on the right half of the real line is called thes-convex function, and it is given as follows.Definition 9.
(see [26]). Let s∈0,1. A function f:0,∞⟶ℝ is said to be an s-convex function in the second sense if(16)fta+1−tb≤tsfa+1−tsfb,holds for all a,b∈0,∞ and t∈0,1.
The notion of them-convex function and strongly m-convex function is defined as follows.Definition 10.
(see [27]). A function f:0,b⟶ℝ is said to be an m-convex function, where m∈0,1 and b>0, if for every x,y∈0,b and t∈0,1, we have(17)ftx+m1−ty≤tfx+m1−tfy.Definition 11.
(see [28]). A function f:0,+∞⟶ℝ is said to be a strongly m-convex function with modulus λ if(18)fta+m1−tb≤tfa+m1−tfb−λmt1−tb−a2,with a,b∈0,+∞ and m∈0,1.
A further generalized convexity is given as follows.Definition 12.
(see [29]). A function f:0,b⟶ℝ is said to be an s,m-convex function, where s,m∈0,12 and b>0, if for every x,y∈0,b and t∈0,1, we have(19)ftx+m1−ty≤tsfx+m1−tsfy.
The notion of the stronglys,m-convex function is defined as follows.Definition 13.
(see [30]). A function f:0,+∞⟶ℝ is said to be a strongly s,m-convex function, with modulus λ≥0, for s,m∈0,12, if(20)fta+m1−tb≤tsfa+m1−tsfb−λt1−tb−a2,holds for all a,b∈0,+∞ and t∈0,1.
Using stronglys,m-convexity and utilizing fractional operators (6) and (7), some fractional integral inequalities are obtained as in [31]. The following result provides the bound of sum of left and right fractional integrals (6) and (7) for strongly s,m-convex functions at an arbitrary point.Theorem 1.
(see [31]). Let f:a,b⟶ℝ be a real-valued function. If f is positive and strongly s,m-convex, then for α,β≥1, the following fractional integral inequality holds:(21)ϵμ,α,l,ω,a+γ,δ,k,cfx;p+ϵμ,β,l,ω,b−γ,δ,k,cfx;p≤fa+mfx/ms+1−λx−ma26m2x−aJα−1,a+x;p+fb+mfx/ms+1−λmb−x26m2b−xJβ−1,b−x;p,x∈a,b.
The following Hadamard-type inequality holds for generalized fractional integral operators for stronglys,m-convex functions.Theorem 2.
(see [31]). Let f:a,b⟶ℝ, a>b, be a real-valued function. If f is positive, strongly s,m-convex and fa+mb−x/m=fx, then for α,β>0, the following fractional integral inequality holds:(22)2s1+mfa+mb2Jα+1,a+b;p+Jβ+1,b−a;p+λ4mb−aβ+2Jβ+1,b−a;p−21+mb−aβ+1Jβ+2,b−a;p+21+m2Jβ+3,b−a;p+b−aα+2×Jα+1,a+b;p−21+mb−aα+1Jα+2,a+b;p+21+m2Jα+3,a+b;p≤ϵμ,α+1,l,ω,a+γ,δ,k,cfb;p+ϵμ,β+1,l,ω,b−γ,δ,k,cfa;p≤Jβ,b−a;p+Jα,a+b;pb−afb+mfa/ms+1−λmb−a26m2.
In the following, using the stronglys,m-convexity of f′, a modulus inequality is obtained.Theorem 3.
(see [31]). Let f:a,b⟶ℝ be a real-valued function. If f is differentiable and f′ is strongly s,m-convex, then for α,β≥1, the following fractional integral inequality holds:(23)ϵμ,α+1,l,ω,a+γ,δ,k,cfx;p+ϵμ,β+1,l,ω,b−γ,δ,k,cfx;p−Jα−1,a+x;pfa+Jβ−1,b−x;pfb≤f′a+mf′x/ms+1−λx−ma26m2x−aJα−1,a+x;p+f′b+mf′x/ms+1−λmb−x26m2b−xJβ−1,b−x;p,x∈a,b.
In [32], we studied the properties of the kernel given in (13). Here, we are interested in the following property.
P: let g and ϕ/x be increasing functions. Then, for x<t<y, x,y∈a,b, the kernel KxyEμ,α,lγ,δ,k,c,g;ϕ satisfies the following inequality:(24)KtxEμ,α,lγ,δ,k,c,g;ϕg′t≤KyxEμ,α,lγ,δ,k,c,g;ϕg′t.
This can be obtained from the following two straightforward inequalities:(25)ϕgt−gxgt−gxg′t≤ϕgy−gxgy−gxg′t,Eμ,α,lγ,δ,k,cωgt−gxμ;p≤Eμ,α,lγ,δ,k,cωgy−gxμ;p.
The reverse of inequality (13) holds when g and ϕ/x are decreasing.
The upcoming section contains the results for unified integral operators dealing with the bounds of several fractional integral operators in a compact form by utilizing stronglys,m-convex functions. A compact version of the Hadamard inequality is presented, and also a modulus inequality is given for the differentiable function f such that f′ is a strongly s,m-convex function. In the whole paper, we will use(26)Ia,b,g≕1b−a∫abgtdt.


## 2. Main Results

The following result provides the upper bound of unified integral operators.Theorem 4.
Letf:a,mb⟶ℝ, 0≤a<mb, be a positive integrable and strongly s,m-convex function, m≠0. Then, for unified integral operators (11) and (12), the following inequality holds:(27)Fgμ,α,l,a+ϕ,γ,δ,k,cfx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α,lγ,δ,k,c,g;ϕmfxmgx−faga−Γs+1x−asmfxmIsx−ga−faIsa+gx+λx−ma2x−a2Ia,x,Idg−a+xIa,x,g+KbxEμ,β,lγ,δ,k,c,g;ϕfbgb−mfxmgx−Γs+1b−xsfbIsb−gx−mfxmIsx+gb+λmb−x2b−x2Ix,b,Idg−x+bIx,b,g.Proof.
ByP, the following inequalities hold:(28)KxtEμ,α,lγ,δ,k,c,g;ϕg′t≤KxaEμ,α,lγ,δ,k,c,g;ϕg′t,a<t<x,(29)KtxEμ,α,lγ,δ,k,c,g;ϕg′t≤KbxEμ,β,lγ,δ,k,c,g;ϕg′t,x<t<b.
For a stronglys,m-convex function, the following inequalities hold for a<t<x and x<t<b, respectively:(30)ft≤x−tx−asfa+mt−ax−asfxm−λx−tt−ax−ma2m2x−a2,(31)ft≤t−xb−xsfb+mb−tb−xsfxm−λt−xb−tmb−x2m2b−x2.
From (28) and (30), one can have(32)∫axKxtEμ,α,lγ,δ,k,c,g;ϕftdgt≤faKxaEμ,α,lγ,δ,k,c,g;ϕ×∫axx−tx−asdgt+mfxmKxaEμ,α,lγ,δ,k,c,g;ϕ∫axt−ax−asdgt−KxaEμ,α,lγ,δ,k,c,g;ϕλx−ma2x−a2∫axx−tt−adgt,i.e.,(33)Fgμ,α,l,a+ϕ,γ,δ,k,cfx,ω;p≤KxaEμ,α,lγ,δ,k,c,g;ϕmfxmgx−faga−Γs+1x−asmfxmIsx−ga−faIsa+gx+λx−ma2x−a2Ia,x,Idg−a+xIa,x,g.
On the other hand, from (29) and (31), one can have(34)∫xbKtxEμ,β,lγ,δ,k,c,g;ϕftdgt≤fbKbxEμ,β,lγ,δ,k,c,g;ϕ×∫xbt−xb−xsdgt+mfxmKxbEμ,β,lγ,δ,k,c,g;ϕ∫xbb−tb−xsdgt−KxbEμ,α,lγ,δ,k,c,g;ϕλmb−x2m2b−x2∫xbt−xb−tdgt,i.e.,(35)Fgμ,β,l,b−ϕ,γ,δ,k,cfx,ω;p≤KbxEμ,β,lγ,δ,k,c,g;ϕfbgb−mfxmgx−Γs+1b−xsfbIsb−gx−mfxmIsx+gb+λmb−x2b−x2Ix,b,Idg−x+bIx,b,g.
By adding (33) and (35), (27) can be obtained.Corollary 1.
Settingp=ω=0 in (27), we can obtain the following inequality involving fractional integral operators defined in [4]:(36)Fα,a+ϕfx;p+Fβ,b−ϕfx;p≤Kga,x;ϕmfxmgx−faga−Γs+1x−asmfxmIsx−ga−faIsa+gx+λx−ma2x−a2Ia,x,Idg−a+xIa,x,g+Kgx,b;ϕfbgb−mfxmgx−Γs+1b−xsfbIsb−gx−mfxmIsx+gb+λmb−x2b−x2Ix,b,Idg−x+bIx,b,g.Remark 1.
(i)
If we considerλ=0 in (27), then Theorem 3.1 in [32] can be obtained, and for λ>0, we get its refinement(ii)
If we considerϕt=tα and gx=x in (27), then Theorem 1 can be obtained(iii)
If we considers=m=1 in the result of (ii), then Corollary 1 in [31] can be obtained(iv)
If we considerα=β in the result of (ii), then Corollary 3 in [31] can be obtained(v)
If we considerf∈L∞a,b in the result of (ii), then Corollary 5 in [31] can be obtained(vi)
If we considerα=β in the result of (v), then Corollary 7 in [31] can be obtained(vii)
If we considers=1 in the result of (ii), then Corollary 5 in [31] can be obtained(viii)
If we considers,m = (1, 1) in (27), then Theorem 2 in [33] is obtained(ix)
If we considerα=β, λ=0, and s,m = (1, 1) in (27), then Theorem 8 in [23] is obtained(x)
If we considerλ=0 and p=ω=0 in (27), then Theorem 1 in [34] is obtained(xi)
If we considerλ=0, ϕt=Γαtα, p=ω=0, and s,m = (1, 1) in (27), then Theorem 1 in [35] is obtained(xii)
If we considerα=β in the result of (xi), then Corollary 1 in [35] is obtained(xiii)
If we considerλ=0, ϕt=tα, gx=x, and m=1 in (27), then Theorem 2.1 in [36] is obtained(xvi)
If we considerα=β in the result of (xiii), then Corollary 2.1 in [36] is obtained(xv)
If we considerλ=0, ϕt=Γαtα/k/kΓkα, s,m = (1, 1), gx=x, and p=ω=0 in (27), then Theorem 1 in [37] can be obtained(xvi)
If we considerα=β in the result of (xv), then Corollary 1 in [37] can be obtained(xvii)
If we considerλ=0, ϕt=Γαtα, gx=x, p=ω=0, and s,m = (1, 1) in (27), then Theorem 1 in [38] is obtained(xviii)
If we considerα=β in the result of (xvii), then Corollary 1 in [38] can be obtained(xviii)
If we considerα=β=1 and x=aorx=b in the result of (xvii), then Corollary 2 in [38] can be obtained(xix)
If we considerα=β=1 and x=a+b/2 in the result of (xvii), then Corollary 3 in [38] can be obtainedThe following lemma is very helpful in the proof of the upcoming theorem, see [31].Lemma 1.
Letf:a,mb⟶ℝ be a strongly s,m-convex function, 0≤a<mb. If f is fa+mb−x/m=fx, m≠0, then the following inequality holds:(37)fa+mb2≤1+mfx2s−λ4ma+mb−x−mx2.In the literature, many mathematicians have established many types of Hadamard inequalities, and for their generalizations, see [39–42]. This also motivates us to introduce the more generalized forms of Hadamard-type inequalities. So, by the help of the abovementioned lemma, the following result provides generalized Hadamard inequality for strongly s,m-convex functions.Theorem 5.
Under the assumptions of Theorem4, in addition to fx=fa+mb−x/m, the following inequality holds:(38)fa+mb22s1+mFgμ,β,l,a+ϕ,γ,δ,k,c1b,ω;p+λ4mFgμ,β,l,b−ϕ,γ,δ,k,ca+mb−x−mx2a,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,c1a,ω;p+λ4mFgμ,β,l,a+ϕ,γ,δ,k,ca+mb−x−mx2b,ω;p≤Fgμ,β,l,a+ϕ,γ,δ,k,cfb,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cfa,ω;p≤KbaEμ,α,lγ,δ,k,c,g;ϕ+KbaEμ,β,lγ,δ,k,c,g;ϕfbgb−mfamga−Γs+1b−asfbIsb−ga−mfamIsa+gb+λmb−a2b−a2Ia,b,Idg−a+bIa,b,g.Proof.
ByP, the following inequalities hold:(39)KxaEμ,α,lγ,δ,k,c,g;ϕg′x≤KbaEμ,α,lγ,δ,k,c,g;ϕg′x,a<x<b,(40)KbxEμ,β,lγ,δ,k,c,g;ϕg′x≤KbaEμ,α,lγ,δ,k,c,g;ϕg′x,a<x<b.
A stronglys,m-convex function satisfying the following inequalities hold for a<x<b:(41)fx≤x−ab−asfb+mb−xb−asfam−λb−xx−ab−ma2m2b−a2.
From (39) and (41), one can have(42)∫abKxaEμ,α,lγ,δ,k,c,g;ϕfxdgx≤mfamKbaEμ,α,lγ,δ,k,c,g;ϕ∫abb−xb−asdgx+fbKbaEμ,α,lγ,δ,k,c,g;ϕ∫abx−ab−asdgx−KabEμ,α,lγ,δ,k,c,g;ϕλb−ma2m2b−a2∫abx−ab−xdgx.
Further, the aforementioned inequality takes the form which involves Riemann–Liouville fractional integrals in the right-hand side, and thus we have upper bound of the unified left-sided integral operator (2) as follows:(43)Fgμ,α,l,b−ϕ,γ,δ,k,cfa,ω;p≤KbaEμ,α,lγ,δ,k,c,g;ϕfbgb−mfamga−Γs+1b−asfbIsb−ga−mfamIsa+gb+λmb−a2b−a2Ia,b,Idg−a+bIa,b,g.
On the other hand, from (39) and (41), the following inequality holds which involves Riemann–Liouville fractional integrals on the right-hand side and gives the estimate of the integral operator (3):(44)Fgμ,α,l,a+ϕ,γ,δ,k,cfb,ω;p≤KbaEμ,β,lγ,δ,k,c,g;ϕfbgb−mfamga−Γs+1b−asfbIsb−ga−mfamIsa+gb+λmb−a2b−a2Ia,b,Idg−a+bIa,b,g.
By adding (43) and (44), the following inequality can be obtained:(45)Fgμ,α,l,b−ϕ,γ,δ,k,cfa,ω;p+Fgμ,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;ϕfbgb−mfamga−Γs+1b−asfbIsb−ga−mfamIsa+gb+λmb−a2b−a2Ia,b,Idg−a+bIa,b,g.
Multiplying both sides of (37) by KxaEμ,α,lγ,δ,k,c,g;ϕg′x and integrating over a,b, we have(46)fa+b2∫abKxaEμ,α,lγ,δ,k,c,g;ϕdgx≤12s1+m∫abKbaEμ,α,lγ,δ,k,c,g;ϕfxdgx−λ4m∫abKxaEμ,α,lγ,δ,k,c,g;ϕa+mb−x−mx2dgx.
From Definition6, the following inequality is obtained:(47)fa+mb22s1+mFgμ,α,l,b−ϕ,γ,δ,k,c1a,ω;p≤Fgμ,α,l,b−ϕ,γ,δ,k,cfa,ω;p−λ4mFgμ,α,l,b−ϕ,γ,δ,k,ca+mb−x−mx2a,ω;p.
Similarly, multiplying both sides of (37) by KbxEμ,β,lγ,δ,k,c,g;ϕg′x and integrating over a,b, we have(48)fa+mb22s1+mFgμ,β,l,a+ϕ,γ,δ,k,c1b,ω;p≤Fgμ,β,l,a+ϕ,γ,δ,k,cfb,ω;p−λ4mFgμ,β,l,a+ϕ,γ,δ,k,ca+mb−x−mx2b,ω;p.
By adding (47) and (48), the following inequality is obtained:(49)fa+mb22s1+mFgμ,β,l,a+ϕ,γ,δ,k,c1b,ω;p+λ4mFgμ,α,l,b−ϕ,γ,δ,k,ca+mb−x−mx2a,ω;p+Fgμ,α,l,b−ϕ,γ,δ,k,c1a,ω;p+λ4mFgμ,β,l,a+ϕ,γ,δ,k,ca+mb−x−mx2b,ω;p≤Fgμ,β,l,a+ϕ,γ,δ,k,cfb,ω;p+Fgμ,α,l,b−ϕ,γ,δ,k,cfa,ω;p.
Using (45) and (49), inequality (38) can be obtained, which completes the proof.Corollary 2.
Settingp=ω=0 in (38), we can obtain the following inequality involving fractional integral operators defined in [4]:(50)fa+mb22s1+mFβ,a+ϕ1b;p+λ4mFα,b−ϕa+mb−x−mx2a;p+Fα,b−ϕ1a;p+λ4mFβ,a+ϕa+mb−x−mx2b;p≤Fβ,a+ϕfb;p+Fα,b−ϕfa;p≤Kga,b;ϕ+Kga,b;ϕfbgb−mfamga−Γs+1b−asfbIsb−ga−mfamIsa+gb+λmb−a2b−a2Ia,b,Idg−a+bIa,b,g.Remark 2.
(i)
If we considerϕt=tα and gx=x in (38), then Theorem 7 in [31] can be obtained(ii)
If we considerλ=0 in the result of (i), then Theorem 8 in [31] can be obtained(iii)
If we considers,m = (1, 1) in (38), then Theorem 3 in [33] is obtained(iv)
If we considerλ=0 and s,m = (1, 1) in (38), then Theorem 22 in [23] is obtained(v)
If we considerλ=0, ϕt=Γαtα+1, p=ω=0, and s,m = (1, 1) in (38), then Theorem 3 in [35] is obtained(vi)
If we considerα=β in the result of (v), then Corollary 3 in [35] is obtained(vii)
If we considerλ=0, ϕt=tα+1, gx=x, and m=1 in (38), then Theorem 2.4 in [36] is obtained(viii)
If we considerα=β in the result of (vii), then Corollary 2.6 in [36] is obtained(ix)
If we considerλ=0, ϕt=Γαtα/k+1, s,m = (1, 1), gx=x, and p=ω=0 in (38), then Theorem 3 in [37] can be obtained(x)
If we considerα=β in the result of (ix), then Corollary 6 in [37] can be obtained(xi)
If we considerλ=0, ϕt=Γαtα+1, p=ω=0, s,m=1, and gx=x in (38), then Theorem 3 in [38] can be obtained(xii)
If we considerα=β in the result of (xi), then Corollary 6 in [38] can be obtainedTheorem 6.
Letf:a,mb⟶ℝ, 0≤a<mb, be a differential function such that f′ is a strongly s,m-convex function, m≠0. Then, for unified integral operators (11) and (12), the following inequality holds:(51)Fgμ,α,l,a+ϕ,γ,δ,k,cf∗gx,ω;p+Fgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α,lγ,δ,k,c,g;ϕmf′xmgx−f′aga−Γs+1x−asmf′xmIsx−ga−f′aIsa+gx+λx−am2x−a2Ia,x,Idg−a+xIa,x,g+KbxEμ,β,lγ,δ,k,c,g;ϕf′bgb−mf′xmgx−Γs+1b−xsf′bIsb−gx−mf′xmIsx+gb+λmb−x2b−x2Ix,b,Idg−x+bIx,b,g,where(52)Fgμ,α,l,a+ϕ,γ,δ,k,cf∗gx,ω;p≔∫axKxtEμ,α,lγ,δ,k,c,g;ϕf′tdgt,Fgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≔∫xbKtxEμ,α,lγ,δ,k,c,g;ϕf′tdgt.Proof.
For a stronglys,m-convex function f′, the following inequalities hold for a<t<x and x<t<b, respectively:(53)f′t≤x−tx−asf′a+mt−ax−asf′xm−λx−tt−ax−ma2m2x−a2,(54)f′t≤t−xb−xsf′b+mb−tb−xsf′xm−λt−xb−tmb−x2m2b−x2.
From (28) and (54), the following inequality is obtained:(55)Fgμ,α,l,a+ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,α,lγ,δ,k,c,g;ϕmf′xmgx−f′aga−Γs+1x−asmf′xmIsx−a−f′aIsa+gx+λx−am2x−a2Ia,x,Idg−a+xIa,x,g.
Similarly, from (29) and (55), the following inequality is obtained:(56)Fgμ,β,l,b−ϕ,γ,δ,k,cf∗gx,ω;p≤KxaEμ,β,lγ,δ,k,c,g;ϕf′bgb−mf′xmgx−Γs+1b−xs−f′bIsb−gx−mf′xmIsx+gb+λmb−x2b−x2Ix,b,Idg−x+bIx,b,g.
By adding (56) and (57), inequality (52) can be achieved.Corollary 3.
Settingp=ω=0 in (52), we can obtain the following inequality involving fractional integral operators defined in [4]:(57)Fα,a+ϕf∗gx,p+Fβ,b−ϕf∗gx,p≤Kga,x;ϕmf′xmgx−f′aga−Γs+1x−asmf′xmIsx−ga−f′aIsa+gx+λx−am2x−a2Ia,x,Idg−a+xIa,x,g+Kbx,b;ϕf′bgb−mf′xmgx−Γs+1b−xsf′bIsb−gx−mf′xmIsx+gb+λmb−x2b−x2Ix,b,Idg−x+bIx,b,g.Remark 3.
(i)
If we considerλ=0 in (52), then Theorem 3.4 in [32] can be obtained(ii)
If we considerϕt=tα and gx=x in (52), then Theorem 6 in [31] can be obtained(iii)
If we considers=m=1 in the result of (ii), then Corollary 13 in [31] can be obtained(iv)
If we considerα=β in the result of (ii), then Corollary 11 in [31] can be obtained(v)
If we considers,m = (1, 1) in (52), then Theorem 3 in [33] is obtained(vi)
If we considerλ=0 and s,m = (1, 1) in (52), then Theorem 25 in [23] is obtained(vii)
If we considerλ=0 and p=ω=0 in (52), then Theorem 2 in [34] is obtained(viii)
If we considerλ=0, ϕt=Γαtα+1, p=ω=0, and s,m = (1, 1) in (52), then Theorem 2 in [35] is obtained(ix)
If we considerα=β in the result of (viii), then Corollary 2 in [35] is obtained(x)
If we considerλ=0, ϕt=tα, gx=x, and m=1 in (52), then Theorem 2.3 in [36] is obtained(xi)
If we considerα=β in the result of (x), then Corollary 2.5 in [36] is obtained(xii)
If we considerλ=0, ϕt=Γαtα/k+1, s,m = (1, 1), gx=x, and p=ω=0 in (52), then Theorem 2 in [37] can be obtained(xiii)
If we considerα=β in the result of (xii), then Corollary 4 in [37] can be obtained(xiv)
If we considerα=β=k=1 and x=a+b/2 in the result of (xii), then Corollary 5 in [37] can be obtained(xv)
If we considerλ=0, ϕt=Γαtα+1, gx=x, p=ω=0, and s,m = (1, 1) in (52), then Theorem 2 in [38] is obtained(xvi)
If we considerα=β in the result of (xv), then Corollary 5 in [38] can be obtained


## 3. Concluding Remarks

In this paper, bounds of a unified integral operator for stronglys,m-convex functions are studied. The compact form of these bounds lead to further interesting consequences with respect to fractional integrals of various kinds for convex, s,m-convex, m-convex, s-convex, and convex functions. These findings are generalized in nature and give the refinements of many inequalities for unified and fractional integral operators via different types of convex functions.



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