# Midpoint Inequalities via Strong Convexity Using Positive Weighted Symmetry Kernels

Authors: Hengxiao Qi; Waqas Nazeer; Sami Ullah Zakir; Kamsing Nonlaopon
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the present research, we generalize the midpoint inequalities for strongly convex functions in weighted fractional integral settings. Our results generalize many existing results and can be considered as extension of existing results.


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## Body

## 1. Introduction

One of the most interesting research areas of classical analysis is the study of functions and operators, especially convex functions, due to its applications in both integration and differentiation. In the last few years, a great effort has been put to develop new inequalities in convex analysis to deal with the various new applications, since modern problems are modeled by fractional calculus and new applications. So, the classical convexity [1, 2] and its related [3, 4] inequalities are not enough to tackle these ones. The new fractional integral inequalities in convexity are always appreciable. Moreover, the generalized and new mode of convexity is the area of interest for most of researcher of convex analysis [5, 6].A functionf:I⟶R is said to be convex on I, if the inequality holds for all r, s ∈I and ϑ∈0,1,
(1)fϑr+1−ϑs≤⩽ϑfr+1−ϑfs.Many problems may discuss in convexity of sets and functions. In recent year, convexity of sets and functions has been main object of study [7–9]. Some new generalized ideas in this point of view are pseudoconvex function, strongly convex function, quasiconvex, generalized convex function, preinvex functions [10], B-convex function, and invex functions. There are many different fundamental books of convex analysis optimization [11, 12].Fractional calculus [13, 14] is not a new concept in mathematics, and similar discussion and controversy are observe in history by famous mathematician like Jensen, Hermite, H older, and Stolz. However, the subject of fractional calculus from an applied point of view got rapid development last years. Like other fields of mathematics, this also influences the integral inequalities and convex analysis ([15]). As a result, various trends in the result are settled recently. The famous fractional integral operators involve Riemann-Liouville [16], Caputo [17, 18], Hadamard [19], and Caputo Fabrizio [20, 21]. For more details about fractional integral operators, we refer [22–24].The classical Hermite–Hadamard inequality is one of the most well-established inequalities in the theory of convex functions with geometrical interpretation, and it has many applications [25–27]. Recall Hermite-Hadamard-type inequality (simply H-H type inequality) which is given as:Suppose functions:c,d⊂R is convex, and then the inequality
(2)sc+d2≤1d−c∫cdsα1dα1≤sc+Sd2is called the Hermite-Hadamard Inequality.In the present research, we generalize the midpoint inequalities for strongly convex functions in weighted fractional integral settings. Our results generalize many existing results and can be considered as extension of existing results.


## 2. Definitions and Basic Results

Definition 1.
Assume thatj⊆R is an interval and that “a” is a positive integer. If a function s:j⊂R⟶R is strongly convex with modulus a, it is called strongly convex with modulus a.
(3)slβ1+1−lβ2≤lsβ1+11−lsβ2−al1−lβ1−β22
for allβ1,β2∈I and l∈0,1.Adamek expanded on the idea of a strongly convex function. They replaced the nonnegative term with a real-valued nonnegative function and defined it as follows:If a function is strongly convex, it is defined as such.(4)slβ1+1−lβ2≤lsβ1+11−lsβ2−l1−lMβ1−β2for allβ1,β2∈I and l∈0,1. See [5, 28, 29] and references therein for more detail about strongly convex functionality.Definition 2 [30].
Lett:d1,d2⟶0,∞ be a function, Then, we say t is symmetric with respect to d1+d2/2 if
(5)td1+d2−β1=tβ1∀β1∈d1,d2.
With the help of above definition, in [31], Fejér gave, namely, the Hermite-Hadamard-Fejér type inequality.
(6)sd1+d22∫d1d2tβ1dβ1⩽∫d1d2sβ1tβ1dβ1⩽sd1+sd22∫d1d2tβ1dβ1,where t is the integrable function.Definition 3.
SupposeIl+asβ1 and Im−asβ1 are the left- and rigt-sided RL fractional integrals of order a>0 defined by [32]
(7)IRLd1+vsβ1=1Γv∫d1β1β1−lv−1sldl,β1>d1,Id2−vRLsβ1=1Γv∫β1d2l−β1a−1sldl,β1<d2,
Endpoint inequalities were found, namely, the generalized and reformulated forms of H-H and H-H-F inequalities in terms of RL fractional integrals, respectively, in [22, 24].
(8)sd1+d22⩽2v−1Γv+1d2−d1vId1+d2/2+vsd2+Id1+d22−vsd2⩽sd1+sd22,sd1+d22Id1+vtd2+Id2−vtd1⩽Id1+vstd2+Id2−vstv⩽sd1+sd22Id1+vtd2+Id2−vtl,
wheres is the positive convex function, continuous on the closed interval d1,d2 when sβ1∈L1d1,d2 and d1<d2.Definition 4.
Letd1,d2⊆R and σβ1 be an increasing positive and monotone function on the interval d1,d2 with a continuous derivative σ′β1 on the open interval d1,d2. Then, left- and right-sided weighted fractional integral of a function s, according to another function σβ1 on d1,d2, is defined by [25]:
(9)Id1+ψv;σsβ1=ψβ1−1Γv∫d1β1σ′lσβ1−σlv−1slψldlIψd2−v;σsβ1=ψβ1−1Γv∫β1mσ′lσl−σβ1a−1slψldl,a>0,
whereψβ1−1≔1/ψβ1 such that ψβ1≠0.Midpoint inequalities were found, namely, the generalized and reformulated forms of H-H and H-H-F inequalities in terms of RL fractional integrals and weighted fractional integrals with positive weighted symmetric function in a kernel, due to using the midpointd1+m/2 of the interval given by, respectively, in [33, 34].
(10)sd1+d22⩽2a−1Γv+1d2−d1vId1+d22+vsd2+Id1+d22−vsd1⩽sd1+sd22,wheres:d1,d2⟶R is convex and continuous, and
(11)sd1+d22Iσ−1d1+v;σψ∘σσ−1d2+Iσ−1d2−v;σψ∘σσ−1d1⩽ψd2Iσ−1d1+ψ∘σv;σs∘σσ−1d2+ψd1Iψ∘σσ−1d1+d22−v;σs∘σσ−1d1⩽sd1+sd22Iσ−1d1+v;σψ∘σσ−1d2+Iσ−1d2−v;σψ∘σσ−1d1,wheres:d1,d2⟶R is convex and continuous, σβ1 is the monotone increasing function from d1,d2 onto itself with σ′ being continuous on d1,d2, and ψ:d1,d2⟶0,∞ is an integrable function, which is symmetric with respect to d1+d2/2, where d1<d2.The fractional integral operators induced a new diversity in inequality theory. There are many fractional integral operators introduced by different mathematicians having their own characteristics.Lemma 5.
Assume thatψ:d1,d2⟶0,∞ is an integrable function and symmetric with respect to d1+d2/2, d1<d2, then
(i)
For eachl∈0,1, we have(12)ψl2d1+2−l2d2=ψ2−l2d1+l2d2.(ii)
Forv>0, we have(13)Iσ−1d1+d22+v:σψ∘σσ−1d2=Iσ−1d1+d22−v:σψ∘σσ−1d1=12Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1.Theorem 6.
Letd2>d1≥0, and let s:d1,d2⟶R be an L1 strong convex function and ψ:d1,d2⟶R be an integrable, positive, and weighted symmetric function w.r.t d1+d2/2. If, in addition, σ is an increasing and positive function from d1,d2 onto itself such that its derivative σ′β1 is continuous on d1,d2, then for n>0, the following inequalities are valid:
(14)sd1+d22Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1≤ψd2Iσ−1d1+d22+ψ∘σv:σs∘σσ−1d2+ψd1Iψ∘σσ−1d1+d22−v:σs∘σσ−1d1−a2Md1−d2Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1≤sd1+sd22−a2Md1−d2Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1.Proof.
The strong convexity of “s” ond1,d2 gives
(15)sβ1+β22≤sβ1+sβ22−b2Mβ1−β2∀β1,β2∈d1,d2,2sβ1+β22≤sβ1+sβ2−aMβ1−β2∀β1,β2∈d1,d2.
So, forβ1=l/2d1+2−l/2d2 and β2=2−l/2d1+l/2d2, l∈0,1, it follows that
(16)2sd1+d22≤sl2d1+2−l2d2+s2−l2d1+l2d2−aMl−1d1−d2.
Multiplying both sides of above equation (16) by lv−1ψl/2d1+2−l/2d2 and integrating the resulting inequality with respect to l over [0, 1] yield that
(17)2sd1+d22∫01lv−1ψl2d1+2−l2d2dl≤∫01lv−1sl2d1+2−l2d2ψl2d1+2−l2d2dl+∫01lv−1s2−l2d1+l2d2wl2d1+2−l2d2dl−∫01lv−1aMl−1d1−d2ψl2d1+2−l2d2dl,(18)2sd1+d22∫01lv−1ψl2d1+2−l2d2dl≤∫01lv−1sl2d1+2−l2d2ψl2d1+2−l2d2dl+∫01lv−1s2−l2d1+l2d2ψl2d1+2−l2d2dl+aMd1−d2∫01lvψl2d1+2−l2d2dl−∫01lv−1ψl2d1+2−l2d2dl.
From the left hand side of inequality in equation (16), we use (13) to obtain
(19)2v−1Γvd2−d1Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1=∫01lv−1ψl2d1+2−l2d2dll≔2d2−σβ1d2−d1,
which follows that(20)2sd1+d22∫01lv−1ψl2d1+2−l2d2dl=2vΓvd2−d1vsd1+d22×Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1.
From the right hand side of inequality in (16), we use (13) to obtain
(21)∫01lv−1sl2d1+2−l2d2ψl2d1+2−l2d2dl+∫01lv−1s2−l2d1+l2d2ψl2d1+2−l2d2dl+aMd1−d2∫01lvψl2d1+2−l2d2dl−∫01lv−1ψl2d1+2−l2d2dl=2vΓvd2−d1vψd2Iσ−1d1+d22+v:σψ∘σσ−1d2+ψd1Iσ−1d1+d22−v:σψ∘σσ−1d1−a2Md1−d2Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1.
By making use of (20) and (21) in (18), we get the desired result.
On the other hand, we can prove the second inequality of Theorem6 by making use of the strong convexity of “s” to get
(22)sl2d1+2−l2d2+s2−l2d1+l2d2≤sd1−sd2−2l−l2Md1−d2.
Multiplying both sides of above equation (16) by lv−1ψl/2d1+2−l/2d2 and integrating the resulting inequality with respect to l over [0, 1] get
(23)∫01lv−1sl2d1+2−l2d2ψl2d1+2−l2d2+∫01lv−1ψl2d1+2−l2d2s2−l2+l2≤sd1−sd2−2l−l2Md1−d2∫01lv−1ψl2d1+2−l2d2.
Now, by usingψl/2d1+2−l/2d2=ψ2−l/2d1+l/2d2 and (21) in (23), we get
(24)2vΓvd2−d1vIσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1−a2Md1−d2Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1≤sd1+sd22−a2Md1−d2Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1This ends our proof.Remark 7.
From Theorem6, we can obtain some special cases as follows:
(1)
Ifa=0, σβ1=x, then inequality (14) becomes(25)sd1+d22Id1+d22+vψd2+Id1+d2/2−vψd2≤ψd2Id1+d22+ψvsd2+ψd1Iψ−d1+d22−vsd1≤sd1+sd22Id1+d22+vψd2+Id1+d22−vψd1,where
(26)Id1+ψvsx=ψ−1xΓv∫d1xx−lv−1slψldl,Iψ−d2vsx=ψ−1xΓv∫xd2l−xv−1slψldlv>0.Lemma 8.
Letd2≥d1>0, let s:d1,d2⟶R be a continuous with a derivative s′∈L1d1,d2 such that sβ1=sd1+∫d1β1s′ldl, and let ψ:d1,d2⟶R be an integrable, positive, and weighted symmetric function with respect to d1+d2/2. If σ is a continuous increasing mapping from the interval d1,d2 onto itself with a derivative σ′β1 which is continuous on d1,d2, then for n>0, the following equality is valid:
(27)sd1+d22Iσ−1d1+d22+v:σψ∘σσ−1d2+Iσ−1d1+d22−v:σψ∘σσ−1d1−ψd2Iσ−1d1+d22+ψ∘σv:σs∘σσ−1d2+ψd1Iψ∘σσ−1d1+d22−v:σs∘σσ−1d1=1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1ψ∘σβ1dβ1s′∘σlσ′ldl−1Γv∫σ−1d1+d22σ−1d2∫σ−1d1lσ′β1d2−σβ1v−1ψ∘σβ1dβ1s′∘σlσ′ldl.


## 3. Main Results

In this section, by using Lemma8, one can extend to some new H-H-F type inequalities for strong convex functions.Theorem 9.
Let0⩽d1<d2, let s:d1,d2⊂0,∞⟶R be a (continuously) differentiable mapping on d1,d2 such that sβ1=sd1+∫dd1β1s′ldl, and let ψ:d1,d2⟶R be an integrable, positive, and weighted symmetric function with respect to d1+d2/2. If in addition, s′ is strong convex on d1,d2, and σ is an increasing and positive function from d1,d2 onto itself such that its derivative σ′β1 is continuous on d1,d2; then for v>0, the following inequalities hold:
(28)Ξ1+Ξ2=1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1ψ∘σβ1dβ1s′∘σlσ′ldl−1Γv∫σ−1d1+d22σ−2d2∫σ−1d1lσ′β1d2−σβ1v−1ψ∘σβ1dβ1s′∘σlσ′ldl≤d2−d1v+12v+2Γv+3ψd1,d1+d22,∞v+3s′d1+v+1s′d2−av+4d2−d122+ψd1+d2,2,d2,∞v+1s′d1+v+3s′d2−av+4d2−d122v≤d2−d1v+1ψd1,d2,∞2v+1Γv+2s′d1+s′d2−ad1−d22.Proof.
By making use of Lemma8 and properties of the modulus, we obtain
(29)Ξ1+Ξ2=1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1ψ∘σβ1dβ1s′∘σlσ′ldl−1Γv∫σ−1d1+d22σ−2d2∫σ−1d1lσ′β1d2−σβ1v−1ψ∘σβ1dβ1s′∘σlσ′ldl≤1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1ψ∘σβ1dβ1s′∘σlσ′ldl−1Γv∫σ−1d1+d22σ−2d2∫σ−1d1lσ′β1d2−σβ1v−1ψ∘σβ1dβ1s′∘σlσ′ldl.
Since∣s′∣ is strongly convex on d1,d2, we get for l∈σ−1d1,σ−1d2:
(30)s′∘σl=s′d2−σld2−d1d1+σl−d2d2−d1d2≤d2−σld2−d1s′d1+σl−d2d2−d1s′d2−ad2−σlσl−d2.
Hence, we obtain(31)Ξ1+Ξ2≤ψd1,d1+d2/2,∞d2−d1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1dβ1×d2−σls′d1+σl−d1s′d2−ad2−σlσl−d2σ′ldl+ψd1+d2/2,d1,∞d2−d1Γv∫σ−1d1+d22σ−1d2∫lσ−1d2σ′β1d2−σβ1v−1dβ1×d2−σls′d1+σl−d1s′d2−ad2−σlσl−d2σ′ldl
By putting value of integration in above inequalities,(32)∫σ−1d1lσ′β1σβ1−d1v−1dβ1=σl−d1vv,∫lσ−1d2σ′β1d2−σvv−1dβ1=d2−σlvv,∫σ−1d1σ−1d1+d22σl−d1v+1σ′ldl=∫σ−1d1+d22σ−1d2d2−σlv+1σ′ldl=d2−d1v+22v+2v+2,∫σ−1d1σ−1d1+d22σl−d1vd2−σlσ′ldl=∫σ−1d1+d22σ−1d2d2−σlvσl−d1σ′ldl=d2−d1v+2v+32v+2v+1v+2,∫σ−1d1σ−1d1+d22σl−d1vv+1d2−σlσ′ldl=d2−d1v+3v+42v+3v+2v+3v,and simple computations yield
(33)=d2−d1v+12v+2Γv+3ψd1,d1+d22,∞v+3s′d1+v+1s′d2−av+4d2−d122+ψd1+d22,d2,∞v+1s′d1+v+3s′d2−av+4d2−d122v≤d2−d1v+1ψd1,d2,∞2v+1Γv+2s′d1+s′d2−ad1−d22,which complete our proof.Remark 10. From Theorem9, we can obtain some special cases as follows:
(1)
Ifσβ1=x,a=0, then inequlaity (28) 3.1 becomes(34)sd1+d22Ivd1+d22+ψd2+Id1+d22−vψd1−ψd2Iψvd1+d22+sd1+ψd2Iwd1+d22−vsd2≤d2−d1v+12v+2Γv+3ψd1,d1+d22,∞v+3s′d1+v+1s′d2+ψd1+d22,d2,∞v+1s′d1+v+3s′d2≤d2−d1v+1ψd1,d2,∞2v+1Γv+2s′d1+s′d2.(2)
Ifa=0, σβ1=x, and ψβ1=1,then inequality 3.1 becomes(35)2v−1Γv+1d2−d1vId1+d22+vψd2+Id1+d22−vψd1−sd1+d22≤d2−d1v+12v+2Γv+3v+3s′d1+v+1s′d2+v+1s′d1+v+3s′d2≤d2−d1v+12v+2Γv+2s′d1+s′d2,(3)
Ifa=0, σβ1=x, ψβ1=1, and v =1, then inequality 3.1 becomes(36)1d2−d1∫d1d2sβ1dβ1−sd1+d22≤c2−c18s′d1+s′d2,Theorem 10.
Let0⩽d1<d2, let s:d1,d2⊂0,∞⟶R be a (continuously) differentiable mapping on d1,d2 such that sβ1=sd1+∫d1β1s′ldl, and let ψ:d1,d2⟶R be an integrable, positive, and weighted symmetric function with respect to d1+d2/2. If, in addition, s′q is strong convex on d1,d2 with q≥1, and “s” is an increasing and positive function from d1,d2 onto itself such that its derivative s′β1 is continuous on d1,d2, then for v>0, the following inequalities hold:
(37)Ξ1+Ξ2≤d2−d1v+12v+1+1/qv+21/qΓv+2×ψd1,d1+d22,∞v+3s′d1q+v+1s′d2q−av+4d2−d1221/q+ψd1+d22,d2,∞v+1s′d1q+v+3s′d2q−av+4d2−d122v1/q≤d2−d1v+1ψd1,d2,∞2v+1+1/qv+21/qΓv+2v+3s′d1q+v+1s′d2q−av+4d2−d1221/q+v+1s′d1q+v+3s′d2q−av+4d2−d1221/q.Proof.
Sinces′q is strong convex on d1,d2, we get for l∈σ−1d1,σ−1d2(38)s′∘σlq=s′d2−eld2−d1d1+σl−d1d2−d1d2q≤d2−σld2−d1s′d1q+σl−d1d2−d1s′d2q−ad2−σlσl−d1.
By making use Lemma 8, power mean inequality, and strong convexity ofs′q, we get
(39)Ξ1+Ξ2v≤1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1ψ∘σβ1dβ1s′∘σlσ′ldl+1Γv∫σ−1d1+d22σ−1d2∫lσ−1d2σ′β1d2−σβ1v−1ψ∘σβ1dβ1s′∘σlσ′ldl≤1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σv−d1v−1ψ∘σβ1dβ1σ′ldl1−1q×1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1ψ∘σvdβ1s′∘σlqσ′ldl1q+1Γv∫σ−1d1+d22v−1d2∫lσ−1d2σ′β1d2−σβ1v−1ψ∘σβ1dvσ′ldl1−1q×1Γv∫σ−1d1+d22σ−1d2∫lσ−1d2σ′β1d2−σβ1v−1ψ∘σβ1dβ1s′∘σlqσ′ldl1q≤ψd1,d1+d2/2,∞Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1dβ1σ′ldl1−1/q×∫σ−1d1σ−1d1+d22∫v−1d1lσ′vσβ1−d1v−1dβ1s′∘σlqσ′ldl1/q+ψd1+d2/2,v,∞Γv∫σ−1d1+v2σ−1d2∫lσ−1d2σ′vd2−σβ1v−1dβ1σ′ldl1−1q×∫σ−1d1+d22v−1d2∫lσ−1d2σ′β1d2−σβ1v−1dβ1s′∘σlqσ′ldl≤ψd1,d1+d2/2,∞Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1dβ1σ′ldl1−1/q×∫v−1d1σ−1d1+v2∫σ−1d1lσ′β1σβ1−d1v−1dβ1×d2−σld2−d1s′d1q+σl−d2d2−d1s′d2q−ad2−σlσl−d2σ′ldl1/q+ψd1+v/2,v,∞Γv∫σ−1d1+d22v−1d2∫lσ−1vσ′β1v−σβ1v−1dβ1σ′ldl1−1/q×∫σ−1d1+v2σ−1v∫lσ−1vσ′β1v−σvv−1dβ1d2−σld2−d1s′d1q+σl−d2d2−d1s′d2q−ad2−σlσl−d2σ′ldl=d2−d1v+12v+1+1/qv+21/qΓv+2ψd1,d1+v2,∞v+3s′d1q+v+1s′vq−av+4d2−d1221/q+ψd1+d22,d2,∞v+1s′d1q+v+3s′d2q−av+4d2−d122v1/q≤d2−d1v+1ψd1,d2,∞2v+1+1/qv+21/qΓv+2v+3s′d1q+v+1s′d2q−av+4d2−d1221/q+v+1s′d1q+v+3s′d2q−av+4d2−d1221/qHence, the proof is completed.Remark 11.
From Theorem10, we can obtain some special cases as follows:
(1)
Theorem 4 of [32] is obtained if we takea=0, ω=ψ, and ρ=σ in Theorem 10(2)
Ifa=0, σβ1=x, then inequality (37) becomes(40)sd1+d22Id1+d22+vψd2+Id1+d22−vψd1−ψd2d1+d22Iψvsd1+ψd2wId1+d22−vsd2≤d2−d1v+12v+1+1/qv+21/pΓv+2ψd1,d1+d22,∞v+3s′d1q+v+1s′d2q1/q+ψd1+d22,d2,∞v+1s′d1q+v+3s′d2q1/q≤d2−d1v+1ψd1,d2,∞2v+1+1/qv+21/qΓv+2v+3s′d1q+v+1s′d2q1/q+v+1s′d1q+v+3s′d2q(3)
Ifa=0, σβ1=x, and ψβ1=1, then inequality (37) becomes(41)∣2v−1Γv+1d2−d1vd1+d22+Ivψd2+Id1+d22−vψd1−sd1+d22∣≤d2−d1v+1ψd1,d2,∞2v+1+2/qpv+11/pΓv+13s′d1q+s′d2q1/q+s′d1q+3s′d2q1/qwhich is already obtained in ((39) [Theorem 5])
(4)
Ifa =0, σβ1=x, ψβ1=1 and v =1, then inequality (37) 3.10 become(42)∣1d2−d1∫d1d2sβ1dβ1−sd1+d22∣≤c2−c183qs′d1q+2s′d2q1/q+2s′d1q+s′d2q1/qTheorem 12.
Let0⩽d1<d2 and s:d1,d2⊂0,∞⟶R be a (continuously) differentiable mapping on d1,d2 such that sβ1=sd1+∫d1β1s′ldl, and let h:d1,d2⟶R be an integrable, positive, and weighted symmetric function with respect to d1+d2/2. If, in addition, s′r is strong convex on d1,d2with 1/v+1/r=1 and r>1, and “s” is an increasing and positive function from d1,d2 onto itself such that its derivative s′β1 is continuous on d1,d2, then for a>0, the following inequalities hold:
(43)Ξ1+Ξ2≤d2−d1v+12v+1+2/qpv+11/pΓv+1×ψd1,d1+d22,∞3s′d1q+s′d2q−av+4d2−d1221/q+vd1+d22,d2,∞v+1s′d1q+v+3s′d2q−av+4d2−d122v1/q≤d2−d1v+1ψd1,d2,∞2v+1+1/qv+21/qΓv+2v+3s′d1q+v+1s′d2q−av+4d2−d1221/q+v+1s′d1q+v+3s′d2q−av+4d2−d1221q.Proof.
Sincess′q is strong convex on d1,d2, we get for l∈σ−1d1,σ−1d2(44)s′∘σlq=s′d2−σld2−d1d1+σl−d1d2−d1d2q≤d2−σld2−d1s′d1q+σl−d1d2−d1s′d2q−ad2−σlσl−d1.
By making use Lemma 8, Hölder inequality, and strong convexity ofs′q and properties of modulus, we have
(45)Ξ1+Ξ2≤1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1ψ∘σβ1dβ1s′∘σlσ′ldl+1Γv∫σ−1d1+d22σ−1d2∫lσ−1d2σ′β1d2−σβ1v−1ψ∘σβ1dβ1s′∘σlσ′ldl≤1Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σv−d1v−1ψ∘σβ1dβ1pσ′ldl1/p×∫σ−1d1σ−1d1+d22s′∘σlqσ′ldl1/q+1Γv∫σ−1d1+d22σ−1d2∫lσ−1d2σ′β1d2−σβ1v−1ψ∘σβ1dβ1pσ′ldl1/p×∫σ−1d1+d22σ−1d2s′∘σlqσ′ldl1/q≤ψd1,d1+d2/2,∞Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σβ1−d1v−1dβ1pσ′ldl1/p×∫σ−1d1σ−1d1+d22s′∘σlqσ′ldl1/q+ψd1+d2/2,d2,∞Γv∫σ−1d1+d22σ−1d2∫lσ−1d2σ′β1d2−σβ1v−1dβ1pσ′ldl1/p×∫σ−1d1+d22σ−1d2s′∘σlqσ′ldl1/q≤ψd1,d1+d2/2,∞Γv∫σ−1d1σ−1d1+d22∫σ−1d1lσ′β1σv−d1v−1dβ1pσ′ldl1/p×∫σ−1d1σ−1d1+d22d2−σld2−d1s′d1q+σl−d2d2−d1s′d2q−ad2−σlσl−d2σ′ldl1/q+ψd1+d2/2,d2,∞Γv∫σ−1d1+d22σ−1d2∫lσ−1d2σ′β1d2−σβ1v−1dβ1pσ′ldl1/p×∫σ−1d1+d22σ−1d2d2−σld2−d1s′d1q+σl−d2d2−d1s′d2q−ad2−σlσl−d2σ′ldl1/q=d2−d1v+12v+1+2/qpv+21/pΓv+1ψd1,d1+d12,∞3s′d1q+s′d2q−2avd2−d1+ψd1+d22,d2,∞s′d1q+3s′d2q−2ad2−d12v1/q≤d2−d1v+1ψd1,d2,∞2v+1+2/qpv+11/pΓv+13s′d1q+s′d2q−2ad2−d121/q+v+1s′d1q+v+3s′d2q−2ad2−d121/q
Hence, the proof is completed.Remark 13.
From Theorem12, we can obtain some special cases as follows:
(1)
Theorem 4 of [32] is obtained if we take a=0, ω=ψ, and ρ=σ in Theorem 12(2)
Ifa=0, σβ1=x, inequality (43) becomes(46)sd1+d22Id1+d22+vψd2+Id1+d22−vψd1−ψd2Id1+d22ψvsd1+ψd2Iwd1+d22−vsd2≤d2−d1v+12v+1+2/qpv+11/pΓv+1ψd1,d1+d22,∞3s′d1q+s′d2q1/q+ψd1+d22,d2,∞s′d1q+3s′d2q−1/q≤d2−d1v+1ψd1,d2,∞2v+1+1/qpv+21/qΓv+23s′d1q+s′d2q1/q+s′d1q+3s′d2q.(3)
Ifa=0, σβ1=ρx=x, and ψβ1=ωx=1, then inequality (43) becomes(47)2v−1Γv+1d2−d1vId1+d22+vψd2+Id1+d22−vψd1−sd1+d22≤d2−d1v+1ψd1,d2,∞2v+1+2/qpv+11/pΓv+13s′d1q+s′d2q1/q+s′d1q+3s′d2q1q,which is already obtained in ((39) [Theorem 6])
(4)
Ifa=0, σβ1=ρx=x, ψβ1=ωx=1, and v=1, then inequality (43) becomes(48)1d2−d1∫d1d2sβ1dβ1−sd1+d22≤c2−c1164p+11p3s′d1q+s′d2q1/q+s′d1q+3s′d2q1q,which is already obtained.


## 4. Conclusions

In this paper, we established the midpoint type inequalities for the strong convex function by using positive weighted symmetry kernels. As an application, our established inequalities can be applied to the special means of real numbers. Our results can be used to estimate error for the midpoint formula. It is interesting to establish midpoint type inequalities for the strong convex function in the setting of a different version of fractional integral operators.



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*Source: 9653481-2021-10-28.xml*