# New Weighted Hermite–Hadamard Type Inequalities for Differentiableh-Convex and Quasi h-Convex Mappings

Authors: Muhammad Amer Latif
Journal: Journal of Mathematics
 (2021)

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## Abstract

In this paper, new weighted Hermite–Hadamard type inequalities for differentiableh-convex and quasi h-convex functions are proved. These results generalize many results proved in earlier works for these classes of functions. Applications of some of our results to s˘-divergence and to statistics are given.


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## Body

## 1. Introduction

The theory of convex functions is based on convex functions stated as follows.A functionp:∅≠K⊆ℜ⟶ℜ is said to be convex on a convex set K if the inequality given as follows:(1)peξ1+1−eξ2≤epξ1+1−epξ2,holds for all ξ1, ξ2∈K and e∈0,1. If (1) holds in reverse direction, then p is said to be concave.The inequality which can be considered as the necessary and sufficient condition of a functionp:∅≠K⊆ℜ⟶ℜ to be convex on n1,n2 is given by [1](2)pn1+n22≤1n2−n1∫n1n2pξ1dξ1≤pn1+pn22,where n1, n2∈K with n1<n2.Inequality (2) is known as Hermite–Hadamard inequality, and it holds in reversed direction if the function p is concave on n1,n2.Over the past three decades, the definition of convex functions and inequality (2) has been subjected to immense research. The definition of convex functions has been modified in various forms, and hence a number of different weighted and nonweighted forms of inequality (2) have been obtained by many researchers.Kirmachi [2] obtained the following estimate for pn1+n2/2−1/n2−n1∫n1n2pξ1dξ1.Theorem 1.
(see [2]). Let p:K⊆ℜ⟶ℜ be a differentiable mapping on K°; let n1,n2∈K° with n1<n2 and p∈Ln1,n2. If p′ is convex on n1,n2, then(3)pn1+n22−1n2−n1∫n1n2pξ1dξ1≤n2−n1p′n1+p′n28.
Pearce and Pecaric [3] improved this estimate by proving the following result.Theorem 2.
(see [3]). Let p:K⊆ℜ⟶ℜ be a differentiable mapping on K°, let n1,n2∈K° with n1<n2, and let q≥1. If p∈Ln1,n2 and p′q is convex on n1,n2, then(4)pn1+n22−1n2−n1∫n1n2pξ1dξ1≤n2−n14p′n1q+p′n2q21/q.
The weighted version of the results in Theorems1 and 2 was obtained in [4].Theorem 3.
(see [4]). Let p:K⊆ℜ⟶ℜ be a differentiable mapping on K°, let n1,n2∈K° with n1<n2, and let g:n1,n2⟶0,∞ be a continuous positive mapping symmetric with respect to n1+n2/2. If pg∈Ln1,n2 and p′ is convex on n1,n2, then(5)pn1+n22∫n1n2gξ1dξ1−∫n1n2pξ1gξ1dξ1≤n2−n1p′n1+p′n22∫01Mg;n1,n2,ede,where Mg;n1,n2,e=∫n1ξn1,n2egξ1dξ1 and ξn1,n2e=1+e/2n1+1−e/2n2.Theorem 4.
(see [4]). Let p:K⊆ℜ⟶ℜ be a differentiable mapping on K°, let n1,n2∈K° with n1<n2, and let g:n1,n2⟶0,∞ be a continuous positive mapping symmetric with respect to n1+n2/2. If pg∈Ln1,n2 and p′q is convex on n1,n2 for q≥1, then(6)pn1+n22∫n1n2gξ1dξ1−∫n1n2pξ1gξ1dξ1≤n2−n1p′n1q+p′n2q21/q∫01Mg;n1,n2,ede,where Mg;n1,n2,e and ξn1,n2e are as defined in Theorem 3.
Under the assumptions of Theorem2, a bound of pn1+pn2/2−1/n2−n1∫n1n2pξ1dξ1 was proposed by Pearce and Pecaric in [3].Theorem 5.
(see [3]). Let p:K⊆ℜ⟶ℜ be a differentiable mapping on K°, let n1,n2∈K° with n1<n2, and let q≥1. If p∈Ln1,n2 and p′q is convex on n1,n2, then(7)pn1+pn22−1n2−n1∫n1n2pξ1dξ1≤n2−n14p′n1q+p′n2q21/q.
The bound of the result of Theorem5 in weighed form was given by Hwang in [5].Theorem 6.
(see [5]). Let p:K⊆ℜ⟶ℜ be a differentiable mapping on K°, let n1,n2∈K° with n1<n2, and let g:n1,n2⟶0,∞ be a continuous positive mapping symmetric with respect to n1+n2/2. If pg∈Ln1,n2 and p′q is convex on n1,n2 for q≥1, then(8)pn1+pn22∫n1n2gξ1dξ1−∫n1n2pξ1gξ1dξ1≤n2−n12p′n1q+p′n2q21/q∫01∫ξn1,n2eηn1,n2egξ1dξ1,where ξn1,n2e=1+e/2n1+1−e/2n2 and ηn1,n2e=1−e/2n1+1+e/2n2.
The concept of quasiconvex functions generalizes the concept of convex functions.Definition 1.
(see [6]). A function p:∅≠K⊆ℜ⟶ℜ is said quasiconvex on K if(9)peξ1+1−eξ2≤maxpξ1,pξ2,holds for all ξ1,ξ2∈K and e∈0,1.
There are quasiconvex functions which are not convex functions (see, for example, [6]).
Alomari et al. [7] obtained the bound of the result of Theorem 5 by using the quasiconvexity of the differentiable mappings.Theorem 7.
(see [7]). Let p:K⊆ℜ⟶ℜ be a differentiable mapping on K°, and let n1,n2∈K° with n1<n2. If p∈Ln1,n2 and p′ is quasiconvex on n1,n2, then(10)pn1+pn22−1n2−n1∫n1n2pξ1dξ1≤n2−n18supp′n1,p′n1+n22+supp′n1+n22,p′n2.
A general form of the result of Theorem7 has been proved by Hwang in [5].Theorem 8.
(see [5]). Under the assumptions of Theorem 6, if p′ is quasiconvex on n1,n2, then(11)pn1+pn22∫n1n2gξ1dξ1−∫n1n2pξ1gξ1dξ1≤n2−n14supp′n1,p′n1+n22+supp′n1+n22,p′n2∫01∫ξn1,n2eηn1,n2egξ1dξ1,where ξn1,n2e and ηn1,n2e are as defined in Theorem 6.
Gavrea [8] extended inequality (10) to weighted form and generalized inequalities (5) and (6) in such a way that the weight function gξ1 is not necessarily symmetric with respect to the midpoint n1+n2/2.
Varošanec [9] generalized the concept of convex functions by giving the concept of h-convex functions.Definition 2.
LetK and J be intervals in ℜ with 0,1⊇J and h:J⟶ℜ be a nonnegative function, where h≢0. A p:K⟶ℜ is an h-convex function or that p belongs to the class SXh,K if p is nonnegative, and for all ξ1,ξ2∈K, e∈0,1, the inequality(12)peξ1+1−eξ2≤hepξ1+h1−epξ2,holds. If inequality (12) is reversed, then p is said to be h-concave or p is said to belong to the class SVh,K.
The classSXh,K of h-convex functions contains all nonnegative convex functions, s-convex functions in the second sense [10], Godunova–Levin functions [11], s -Godunova–Levin type, tgs-convex, and P-functions [12] as special cases.
Inspired by the research towards this direction, the main objectives of this paper are to introduce the notion of quasih-convex functions and to acquire new weighted Hermite–Hadamard type inequalities for h -convex and quasi h-convex mappings. The results of this paper generalize the results of Gavrea [8] and in particular contain the results for all nonnegative convex functions, s-convex functions, Godunova–Levin functions, s-Godunova–Levin functions, tgs-convex, quasi convex functions, and P-functions.
In Section2, we recall some integral identities for a differentiable mapping and a symmetric function with respect to n1+n2/2 defined over an interval n1,n2. In Section 2, an important inequality for positive linear functional on Cn1,n2 and an h-convex function is proved to obtain some very stimulating results of this manuscript. Section 3 contains some new weighted Hermite–Hadamard type integral inequalities related with the left and right parts of Hermite–Hadamard inequalities (2). The results of Section 3 provide weighted generalization of a number of results proved so far in the field of mathematical inequalities for differentiable h-convex and quasi h-convex functions [13–22].


## 2. Some Auxiliary Results

The following notations and results have been used in [8].Letr:n1,n2⟶0,∞ be a continuous function with(13)∫n1n2rξ1dξ1=1,and the integral ∫n1n2ξ1rξ1dξ1 is denoted by α, that is,(14)α=∫n1n2ξ1rξ1dξ1.In case, whenrξ1:n1,n2⟶0,∞ is symmetric with respect to n1+n2/2, that is, if(15)rn1+n2−ξ1=rξ1,then the following result holds.Lemma 1.
(see [8]). If rξ1:n1,n2⟶0,∞ is symmetric with respect to n1+n2/2, then(16)α=n1+n22.
Now, we introduce the notion of the quasih-convex functions as follows.Definition 3.
LetK and J be intervals in ℜ with 0,1⊇J and h:J⟶ℜ be a nonnegative function, where h≢0. A p:K⟶ℜ is an quasi h-convex function, or that p belongs to the class SQh,K if p is nonnegative, and for all ξ1,ξ2∈K, e∈0,1, the inequality(17)peξ1+1−eξ2≤supλpξ1,μpξ2,holds, where λ=supe∈0,1he and μ=supe∈0,1h1−e. If inequality (17) is reversed, then p is said to be quasi h-concave or p is said to belong to the class SQ′h,K.Example 1.
Consider the functionp:−2,2⟶ℜ defined as(18)pe=1,e∈−2,−1,e2,e∈−1,2,hn2e=en2,n2>0,0<e<1.
Then,p is quasi hn2-convex but not hn2-convex on −2,2.
Now onwards, we suppose thatκn1,n2ξ1=n2−ξ1/n2−n1 and ηn1,n2ξ2=1−ξ2/2n1+1+ξ2/2n2.Lemma 2.
Letp:K⊆ℜ⟶ℜ be a differentiable mapping on K° and p′∈Ln1,n2, where n1,n2⊆K°. Let r:n1,n2⟶0,∞ be a continuous mapping and h:J⊇0,1⟶ℜ be a real nonnegative function, such that h≢0. Then,(19)pn1∫n1n2hκn1,n2ξ1rξ1n2ξ1+pn2∫n1n2h1−κn1,n2ξ1rξ1dξ1−∫n1n2hκn1,n2ξ1+h1−κn1,n2ξ1pξ1rξ1dξ1=α−n1∫01Jr,n1,α;ξ2p′ξ2n1+1−ξ2αdξ2+n2−α∫01Jr,α,n2;ξ2p′1−ξ2α+ξ2n2dξ2,where(20)Jr,α,β;ξ2=∫n11−ξ2α+ξ2βh1−κn1,n2ξ1rξ1dξ1−∫1−ξ2α+ξ2βn2hκn1,n2ξ1rξ1dξ1,α,β∈n1,n2.Proof.
The following identities hold:(21)pξ1−pn1=∫n1n2σξ1−ep′ede,(22)pξ1−pn2=−∫n1n2σe−ξ1p′ede,where σ⋅ is the Heavyside function defined by(23)σξ1=0,ξ1<0,1,ξ1>0.
Multiplying both sides of (21) with hκn1,n2ξ1rξ1 and integrating over n1,n2, we have(24)∫n1n2hκn1,n2ξ1rξ1pξ1dξ1−pn1∫n1n2hκn1,n2ξ1rξ1dξ1=∫n1n2∫en2hκn1,n2ξ1rξ1dξ1p′ede.
Similarly, multiplying both sides of (22) with h1−κn1,n2ξ1rξ1 and integrating over n1,n2, we also have(25)∫n1n2h1−κn1,n2ξ1rξ1pξ1dξ1−pn2∫n1n2h1−κn1,n2ξ1rξ1dξ1=−∫n1n2∫n1eh1−κn1,n2ξ1rξ1dξ1p′ede.
From (24) and (25), we get(26)pn1∫n1n2hκn1,n2ξ1rξ1dξ1+pn2∫n1n2h1−κn1,n2ξ1rξ1dξ1−∫n1n2hκn1,n2ξ1+h1−κn1,n2ξ1pξ1rξ1dξ1=∫n1n2∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1p′ede=∫n1α∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1n2ξ1p′ede+∫αn2∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1p′ede.
In the last identity, we sete=1−ξ2n1+ξ2α for the first integral and e=1−ξ2α+ξ2n2 for the second integral, and we obtain (19).Remark 1.
If we takehe=e in Lemma 2, then we get the result for nonnegative convex functions similar to that of (see page 94 in Lemma 2.2. in [8]).Corollary 1.
If we takerξ1=1/n2−n1, for all ξ1∈n1,n2, then (19) reduces to(27)pn1+pn2n2−n1∫n1n2h1−κn1,n2ξ1dξ1−1n2−n1∫n1n2hκn1,n2ξ1+h1−κn1,n2ξ1pξ1dξ1=n2−n12×∫01J1n1,n2;ξ2p′ηn1,n2ξ2−p′ηn1,n2n1+n2−ξ2dξ2,where(28)J1n1,n2;ξ2=1n2−n1∫n1n1+n2−ηn1,n2ξ2h1−κn1,n2ξ1dξ1−∫n1+n2−ηn1,n2ξ2n2hκn1,n2ξ1dξ1.Proof.
We know that(29)α=1n2−n1∫n1n2ξ1dξ1=n1+n22.
Hence,(30)Jr,n1,α;ξ2=J1n2−n1,n1,n1+n22;ξ2=1n2−n1∫n1n1+n2−ηn1,n2ξ2h1−κn1,n2ξ1dξ1−1n2−n1∫n1+n2−ηn1,n2ξ2n2hκn1,n2ξ1dξ1=1n2−n1∫ηn1,n2ξ2n2hκn1,n2ξ1dξ1−1n2−n1∫n1ηn1,n2ξ2h1−κn1,n2ξ1dξ1=−J1n2−n1,n1+n22,n2;ξ2=−Jr,α,n2;ξ2.Corollary 2.
If the functionrξ1 is symmetric with respect to n1+n2/2 on n1,n2, then(31)pn1+pn2∫n1n2h1−κn1,n2ξ1rξ1dξ1−∫n1n2hκn1,n2ξ1+h1−κn1,n2ξ1pξ1rξ1dξ1=n2−n12×∫01J2r,n1,n2;ξ2p′ηn1,n2ξ2−p′n1+n2−ηn1,n2ξ2dξ2,where(32)J2r,n1,n2;ξ2=∫n1ηn1,n2ξ2h1−κn1,n2ξ1rξ1dξ1−∫ηn1,n2ξ2n2hκn1,n2ξ1rξ1dξ1.Proof.
Since the functionrξ1 is symmetric with respect to n1+n2/2 on n1,n2, we have(33)∫n1n2rξ1dξ1=1,∫n1n2ξ1rξ1dξ1=n1+n22.
Moreover,(34)Jr,n1,α;ξ2=Jr,n1,n1+n22;ξ2=−Jr,n1+n22,n2;ξ2=−Jr,α,n2;ξ2.
Hence, from (19), we get the required identity (31).
Now, we will discuss some cases for Lemma2.(1)
Ifhe=1, then we have the result for P-functions.Corollary 3.
Under the assumptions of Corollary1, if p is P-function on n1,n2, then(35)pn1+pn2−2n2−n1∫n1n2pξ1dξ1=n2−n12∫01p′ηn1,n2ξ2−p′n1+n2−ηn1,n2ξ2dξ2.(2)
Ifhe=es, then we obtain the following result for s-convex functions.Corollary 4.
Suppose that the conditions of Corollary1 are fulfilled and if p is s-convex function on n1,n2, then(36)pn1+pn2s+1−1n2−n1∫n1n2κn1,n2ξ1s+1−κn1,n2ξ1spξ1dξ1=n2−n12∫01μξ2,sp′ηn1,n2ξ2−p′n1+n2−ηn1,n2ξ2dξ2,where(37)μξ2,s=1−ξ2s+1−1+ξ2s+12s+1s+1.(3)
Ifhe=e−s, e∈0,1, and s∈0,1, then we obtain the result for function of s-Godunova–Levin type.Corollary 5.
Under the assumptions of Corollary1, if p is function of s-Godunova–Levin type on n1,n2, then(38)pn1+pn21−s−1n2−n1∫n1n2κn1,n2ξ1−s+1−κn1,n2ξ1−spξ1dξ1=n2−n12∫01μξ2,−sp′ηn1,n2ξ2−p′n1+n2−ηn1,n2ξ2dξ2,where(39)μξ2,−s=1−ξ21−s−1+ξ21−s21−s1−s,ξ2∈0,1,s∈0,1.(4)
Ifhe=e1−e, e∈0,1, then we obtain the result for egs-convex functions.Corollary 6.
Under the assumptions of Corollary1, if p is result for egs-convex functions on n1,n2, then(40)pn1+pn26−2n2−n1∫n1n21−κn1,n2ξ1κn1,n2ξ1pξ1dξ1=n2−n124∫01ξ2ξ22−3p′ηn1,n2ξ2−p′n1+n2−ηn1,n2ξ2dξ2.Lemma 3.
LetA:Cn1,n2⟶ℜ be a positive linear functional on Cn1,n2, and let ei be monomials eiξ1=ξ1i, ξ1∈n1,n2, i∈ℕ. Let g be a h-convex function on n1,n2, then(41)Age1≤Ahκn1,n2e1gn1+Ah1−κn1,n2e1gn2.Proof.
By using theh-convexity of g on n1,n2 and the given equality(42)e1=κn1,n2e1n1+1−κn1,n2e1n2,we get(43)ge1=gκn1,n2e1n1+1−κn1,n2e1n2≤hκn1,n2e1gn1+h1−κn1,n2e1gn2.
SinceA is a positive linear functional, we get inequality (41) by applying A on both sides of (43).


## 3. Main Results

The following theorem generalizes the result given by Gavrea in [8].Theorem 9.
Letp:K⊆ℜ⟶ℜ be a differentiable mapping on K° and p′∈Ln1,n2, where n1,n2⊆K°. If r:n1,n2⟶0,∞ is a continuous mapping and p′ is h-convex on n1,n2, then the following inequality holds:(44)∫n1n2pξ1rξ1n2ξ1−pα≤Zn1,α,n2p′n1+Gn1,α,n2p′n2,where(45)Zn1,α,n2=∫n1α∫n1erξ1n2ξ1hκn1,n2ede+∫αn2∫en2rξ1n2ξ1hκn1,n2ede,Gn1,α,n2=∫n1α∫n1erξ1n2ξ1h1−κn1,n2ede+∫αn2∫en2rξ1n2ξ1h1−κn1,n2ede.Proof.
We can write(46)pξ1−pα=∫n1n2σξ1−e−σα−ep′ede.
From (46), we obtain(47)∫n1n2pξ1rξ1dξ1−pα=∫n1n2∫en2rξ1dξ1−σα−ep′ede.
Taking absolute value on both sides of (47) and applying Lemma 3, we have(48)∫n1n2pξ1rξ1dξ1−pα≤∫n1n2∫en2rξ1dξ1−σα−ep′ede≤p′n1∫n1n2∫en2rξ1dξ1−σα−ehκn1,n2ede+p′n2∫n1n2∫en2rξ1dξ1−σα−ehκn1,n2ede.
We notice that(49)∫n1n2∫en2rξ1dξ1−σα−ehκn1,n2ede=∫n1α∫en2rξ1dξ1−∫n1n2rξ1dξ1hκn1,n2ede+∫αn2∫en2rξ1dξ1hκn1,n2ede=∫n1α∫n1erξ1dξ1hκn1,n2ede+∫αn2∫en2rξ1dξ1hκn1,n2ede=Zn1,α,n2.
In a similar way,(50)∫n1n2∫en2rξ1dξ1−σα−eh1−κn1,n2ede=∫n1α∫n1erξ1dξ1h1−κn1,n2ede+∫αn2∫en2rξ1dξ1h1−κn1,n2ede=Gn1,α,n2.
We get the result from (49) and (50).Corollary 7.
Suppose that the assumptions of Theorem9 are satisfied and that rξ1 is symmetric with respect to n1+n2/2 on n1,n2, then(51)∫n1n2pξ1rξ1dξ1−pn1+n22≤p′n1+p′n2Z1n1,n2,where(52)Z1n1,n2=∫n1n1+n2/2∫n1erξ1dξ1hκn1,n2ξ1de+∫n1+n2/2n2∫en2rξ1dξ1hκn1,n2ξ1de.Proof.
Since the functionrξ1 is symmetric with respect to n1+n2/2 on n1,n2 so α=n1+n2/2 and the function rξ1 is symmetric with respect n1+n2/2 on n1,n2, this fact gives(53)Zn1,α,n2=Zn1,n1+n22,n2=Gn1,n1+n22,n2=Gn1,α,n2.
Thus,(54)Zn1,n1+n22,n2=Gn1,n1+n22,n2=∫n1n1+n2/2∫n1erξ1dξ1hκn1,n2ede+∫n1+n2/2n2∫en2rξ1dξ1hκn1,n2ede=Z1n1,n2.Corollary 8.
If we takerξ1=gξ1/∫n1n2gξ1dξ1 in (44) and gξ1 is symmetric with respect to n1+n2/2. Then, the following inequality holds:(55)∫n1n2pξ1gξ1dξ1−pn1+n22∫n1n2gξ1dξ1≤p′n1+p′n2Z2n1,n2,Z2n1,n2=∫n1n1+n2/2∫n1egξ1n2ξ1hκn1,n2ede+∫n1+n2/2n2∫en2gξ1dξ1hκn1,n2ede.Theorem 10.
Letp:K⊆ℜ⟶ℜ be a differentiable mapping on K° and p′∈Ln1,n2, where n1,n2⊆K°. If r:n1,n2⟶0,∞ is a continuous mapping and p′q is h-convex on n1,n2 for q≥1, then(56)∫n1n2pξ1rξ1dξ1−pα≤2∫αn2ξ1−αrξ1dξ11−1/q×Zn1,α,n2p′n1q+Gn1,α,n2p′n2q1/q,where Zn1,α,n2 and Gn1,α,n2 are given in Theorem 9.Proof.
Application of Hölder inequality in (47) yields that(57)∫n1n2pξ1rξ1dξ1−pα≤∫n1n2∫en2rξ1dξ1−σα−ep′eqde≤∫n1n2∫en2rξ1dξ1−σα−ede1−1/q×∫n1n2∫en2rξ1dξ1−σα−ep′eqde1/q.
Applying Lemma3, we have(58)∫n1n2∫en2rξ1dξ1−σα−ep′eqde≤Zn1,α,n2p′n1q+Gn1,α,n2p′n2q.
On the other hand, we have(59)∫n1n2∫en2rξ1dξ1−σα−ede=∫n1α∫n1erξ1dξ1de+∫αn2∫en2rξ1dξ1n2e=2∫αn2ξ1−αrξ1dξ1.
A combination of (57)–(59) gives (56).Corollary 9.
Ifrξ1 is symmetric with respect to n1+n2/2 on n1,n2, then from (56), we obtain(60)∫n1n2pξ1rξ1dξ1−pn1+n22≤Z11/qn1,n22∫αn2ξ1−αrξ1dξ11−1/qp′n1q+p′n2q1/q,where Z1n1,n2 is as defined in Corollary 7.Corollary 10.
Ifrξ1=gξ1/∫n1n2gξ1dξ1 and gξ1 is symmetric with respect to n1+n2/2 on n1,n2, then the following inequality holds:(61)∫n1n2pξ1gξ1dξ1−pn1+n22∫n1n2gξ1dξ1≤Z21/qn1,n22∫n1n1+n2/2n1+n22−ξ1rξ1dξ11−1/qp′n1q+p′n2q1/q,where Z2n1,n2 is as defined in Corollary 8.
For our next results, we use the following notations.(62)φr,p≔pn1∫n1n2hκn1,n2ξ1rξ1dξ1+pn2∫n1n2h1−κn1,n2ξ1rξ1dξ1−∫n1n2hκn1,n2ξ1+h1−κn1,n2ξ1pξ1rξ1dξ1.
It is clear from (62) that(63)φ1n2−n1,p≔pn1∫n1n2hκn1,n2ξ1dξ1+pn2∫n1n2h1−κn1,n2ξ1dξ1n2−n1−1n2−n1∫n1n2hκn1,n2ξ1+h1−κn1,n2ξ1pξ1dξ1.
The next result gives upper bound ofφr,p when the function pξ1 is quasi h-convex.Theorem 11.
Letp:K⊆ℜ⟶ℜ be a differentiable mapping on K° and p′∈Ln1,n2, where n1,n2⊆K°. If r:n1,n2⟶0,∞ be a continuous mapping and p′ is quasi h-convex on n1,n2, then the following inequality holds:(64)φr,p≤α−n1supλp′n1,μp′α∫01Jr,n1,α;ξ2dξ2+n2−αsupμp′α,λp′n2∫01Jr,α,n2;ξ2dξ2,where Jr,α,β;ξ2 is defined as in Lemma 2, λ=supe∈0,1he, and μ=supe∈0,1h1−e.Proof.
Sincep′ is quasi h-convex on n1,n2, we have(65)p′ξ2n1+1−ξ2α≤supλp′n1,μp′α,p′1−ξ2α+ξ2n2≤supμp′α,λp′n2,for all ξ2∈0,1. Hence, inequality (64) follows from (19).Theorem 12.
Letp:K⊆ℜ⟶ℜ be a differentiable mapping on K° and p′∈Ln1,n2, where n1,n2⊆K°. If r:n1,n2⟶0,∞ be a continuous mapping and p′ is quasi h-convex on n1,n2, then(66)φr,p≤Z3n1,n2supλp′n1,μp′n1+n22+supλp′n1+n22,μp′n2,where(67)Z3n1,n2=∫n1n1+n2/2∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1de.
λ and μ are defined as in Theorem 11.Proof.
The symmetry ofrξ1 with respect to n1+n2/2 on n1,n2 gives(68)φr,p=pn1+pn2∫n1n2h1−κn1,n2ξ1rξ1dξ1−∫n1n2hκn1,n2ξ1+h1−κn1,n2ξ1pξ1rξ1dξ1.
We also observe that(69)α−n1∫01Jr,n1,α;ξ2dξ2=∫n1α∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1de,n2−α∫01Jr,α,n2;ξ2dξ2=∫αn2∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1de.
Consider the functionp:n1,n2⟶ℜ defined by(70)pe=∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1.
Then,(71)p′e=h1−κn1,n2e+hκn1,n2ere>0,e∈n1,n2.
This shows thatpe is an increasing function on n1,n2 and(72)pn1+n22=0.
Now, it is easy to see that(73)α−n1∫01Jr,n1,α;ξ2dξ2=∫n1n1+n2/2∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1de,n2−α∫01Jr,α,n2;ξ2dξ2=∫n1+n2/2n2∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1de=∫n1n1+n2/2∫n1eh1−κn1,n2ξ1rξ1dξ1−∫en2hκn1,n2ξ1rξ1dξ1de.
Hence, inequality (66) follows from the inequality.Remark 2.
If we choosehe=e in Theorem 9, Corollary 7 and 8, Theorem 10, Corollaries 9 and 10, and Theorems 11 and 12, we get the results for nonnegative convex functions and quasi-convex functions (see Theorem 3.1, Corollary 3.1, Theorem 3.2, Remark 3.2, Theorem 3.3 and Theorem 3.4 in [8]). Moreover, one can obtain inequalities for s-convex functions, Godunova–Levin functions, s-Godunova–Levin functions, egs-convex, P-functions, and quasiconvex functions from the result of Theorem 9, Corollaries 7 and 8, Theorem 10, Corollaries 9 and 10, and Theorems 11 and 12 by choosing he=es, e−1, e−s, e1−e and 1, respectively.


## 4. Applications

### 4.1.s˘-Divergence Measures

Here, we provide some applications ons˘-divergence measure and probability density function by using the results proved in Section 3. Let the set ϕ and the σ finite measure μ be given, and let the set of all probability densities on μ to be defined on Ω≔ξ2|ξ2:ϕ⟶ℜ,ξ2λ>0,∫ϕξ2λdμλ=1. Let s˘:0,∞⟶ℜ be given mapping and consider Dsξ2,ξ1 defined by(74)Dsξ2,ξ1≔∫ϕξ2λs˘ξ1λξ2λdμλ,ξ2,ξ1∈Ω.Ifs˘ is convex, then (74) is called as the Csisźar s˘-divergence. Consider the following Hermite–Hadamard ℋℋ divergence:(75)Dℋℋs˘ξ2,ξ1≔∫ϕξ2λ∫1ξ1λ/ξ2λs˘ξ1dξ1ξ1λ/ξ2λ−1dμλ,ξ2,ξ1∈Ω,where s˘ is convex on 0,∞ with s˘1=0. Note that Dℋℋs˘ξ2,ξ1≥0 with the equality holds if and only if ξ2=ξ1.Proposition 1.
Lets˘:K⊆ℜ⟶ℜ be a differentiable mapping on K°, where n1,n2⊆K°. If rξ1=1/n2−n1, ξ1∈n1,n2, s˘′q is h-convex on n1,n2 for q≥1 with h=1 |No Image for this Article and s˘1=0, then(76)Dℋℋs˘ξ2,ξ1−∫ϕξ2λs˘ξ1λ+ξ2λ2ξ2λdμλ≤∫ϕξ2λξ1λ−ξ2λ4ξ2λs˘′1q+s˘′ξ1λξ2λq1/qdμλ.Proof.
Letϑ1≔λ∈ϕ:ξ1λ>ξ2λ, ϑ2≔λ∈ϕ:ξ1λ<ξ2λ, and ϑ3≔λ∈ϕ:ξ1λ=ξ2λ. Obviously, if λ∈ϑ3, then equality holds in (76). Now, if λ∈ϑ1, then for n1=1 and n2=ξ1λ/ξ2λ in Theorem 10, multiplying both sides to the obtained result by ξ2λ and integrating over ϑ1, we have(77)∫ϑ1ξ2λ∫1ξ1λ/ξ2λs˘ξ1dξ1ξ1λ/ξ2λ−1dμλ−∫ϑ1ξ2λs˘ξ1λ+ξ2λ2ξ2λdμλ≤∫ϑ1ξ2λξ1λ−ξ2λ4ξ2λs˘′1q+s˘′ξ1λξ2λq1/qdμλ.
Similarly, ifλ∈ϑ2, then for n2=1 and n1=ξ1λ/ξ2λ in Theorem 9, multiplying both sides to the obtained result by ξ2λ and integrating over ϑ2, we have(78)∫ϑ2ξ2λ∫1ξ1λ/ξ2λs˘ξ1dξ1ξ1λ/ξ2λ−1dμλ−∫ϑ2ξ2λs˘ξ1λ+ξ2λ2ξ2λdμλ≤∫ϑ2ξ2λ−ξ1λ4ξ2λs˘′ξ1λξ2λq+s˘′1q1/qdμλ.
Adding inequalities (77) and (78) and utilizing triangle inequality, we get the desired result (76).


### 4.2. Applications to Statistics

Letr:n1,n2⟶0,∞ be the probability density function of a continuous random variable X symmetric to n1+n2/2 with 0<n1<n2. The rth moment of X is defined as(79)ErX=∫n1n2ξ1rrξ1dξ1,which is assumed to be finite.Theorem 13.
Suppose that0<n1<n2 and r≥2, then the following inequality holds:(80)ErX−n1+n22r≤rn2−n1n1r−1+n2r−12.Proof.
Letpξ1=ξ1r on n1,n2 for r≥2, and we have p′ξ1=rξ1r−1 is convex on [n1, n2]. Since the functions r:n1,n2⟶0,∞ are symmetric with respect to n1+n2/2,(81)∫n1erξ1dξ1≤∫n1n2rξ1dξ1=1,∫en2rξ1dξ1≤∫n1n2rξ1dξ1=1,α=n1+n22.
Therefore, from inequality (44), we obtain inequality (80).


## 4.1.s˘-Divergence Measures

Here, we provide some applications ons˘-divergence measure and probability density function by using the results proved in Section 3. Let the set ϕ and the σ finite measure μ be given, and let the set of all probability densities on μ to be defined on Ω≔ξ2|ξ2:ϕ⟶ℜ,ξ2λ>0,∫ϕξ2λdμλ=1. Let s˘:0,∞⟶ℜ be given mapping and consider Dsξ2,ξ1 defined by(74)Dsξ2,ξ1≔∫ϕξ2λs˘ξ1λξ2λdμλ,ξ2,ξ1∈Ω.Ifs˘ is convex, then (74) is called as the Csisźar s˘-divergence. Consider the following Hermite–Hadamard ℋℋ divergence:(75)Dℋℋs˘ξ2,ξ1≔∫ϕξ2λ∫1ξ1λ/ξ2λs˘ξ1dξ1ξ1λ/ξ2λ−1dμλ,ξ2,ξ1∈Ω,where s˘ is convex on 0,∞ with s˘1=0. Note that Dℋℋs˘ξ2,ξ1≥0 with the equality holds if and only if ξ2=ξ1.Proposition 1.
Lets˘:K⊆ℜ⟶ℜ be a differentiable mapping on K°, where n1,n2⊆K°. If rξ1=1/n2−n1, ξ1∈n1,n2, s˘′q is h-convex on n1,n2 for q≥1 with h=1 |No Image for this Article and s˘1=0, then(76)Dℋℋs˘ξ2,ξ1−∫ϕξ2λs˘ξ1λ+ξ2λ2ξ2λdμλ≤∫ϕξ2λξ1λ−ξ2λ4ξ2λs˘′1q+s˘′ξ1λξ2λq1/qdμλ.Proof.
Letϑ1≔λ∈ϕ:ξ1λ>ξ2λ, ϑ2≔λ∈ϕ:ξ1λ<ξ2λ, and ϑ3≔λ∈ϕ:ξ1λ=ξ2λ. Obviously, if λ∈ϑ3, then equality holds in (76). Now, if λ∈ϑ1, then for n1=1 and n2=ξ1λ/ξ2λ in Theorem 10, multiplying both sides to the obtained result by ξ2λ and integrating over ϑ1, we have(77)∫ϑ1ξ2λ∫1ξ1λ/ξ2λs˘ξ1dξ1ξ1λ/ξ2λ−1dμλ−∫ϑ1ξ2λs˘ξ1λ+ξ2λ2ξ2λdμλ≤∫ϑ1ξ2λξ1λ−ξ2λ4ξ2λs˘′1q+s˘′ξ1λξ2λq1/qdμλ.
Similarly, ifλ∈ϑ2, then for n2=1 and n1=ξ1λ/ξ2λ in Theorem 9, multiplying both sides to the obtained result by ξ2λ and integrating over ϑ2, we have(78)∫ϑ2ξ2λ∫1ξ1λ/ξ2λs˘ξ1dξ1ξ1λ/ξ2λ−1dμλ−∫ϑ2ξ2λs˘ξ1λ+ξ2λ2ξ2λdμλ≤∫ϑ2ξ2λ−ξ1λ4ξ2λs˘′ξ1λξ2λq+s˘′1q1/qdμλ.
Adding inequalities (77) and (78) and utilizing triangle inequality, we get the desired result (76).


## 4.2. Applications to Statistics

Letr:n1,n2⟶0,∞ be the probability density function of a continuous random variable X symmetric to n1+n2/2 with 0<n1<n2. The rth moment of X is defined as(79)ErX=∫n1n2ξ1rrξ1dξ1,which is assumed to be finite.Theorem 13.
Suppose that0<n1<n2 and r≥2, then the following inequality holds:(80)ErX−n1+n22r≤rn2−n1n1r−1+n2r−12.Proof.
Letpξ1=ξ1r on n1,n2 for r≥2, and we have p′ξ1=rξ1r−1 is convex on [n1, n2]. Since the functions r:n1,n2⟶0,∞ are symmetric with respect to n1+n2/2,(81)∫n1erξ1dξ1≤∫n1n2rξ1dξ1=1,∫en2rξ1dξ1≤∫n1n2rξ1dξ1=1,α=n1+n22.
Therefore, from inequality (44), we obtain inequality (80).


## 5. Conclusion

In this study, we propose new definition, namely, the definition of quasih-convex functions and provide an example of such type of functions. We prove some new weighted Hermite–-Hadamard type inequalities for differentiable for h-convex and quasi h-convex functions when the weight function is not necessarily symmetric about the midpoint of the interval. These results generalize many results proved in earlier works for these classes of functions. Applications of some of our results to s˘-divergence and to statistics are given. We believe that the results of the current study may be a motivation to explore more new results relevant to this field of research for people working in the rich field of mathematical inequalities.



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