# Hermite–Hadamard-Type Inequalities for Product of Functions by Using Convex Functions

Authors: Tariq Nawaz; M. Asif Memon; Kavikumar Jacob
Journal: Journal of Mathematics
 (2021)

---

## Abstract

One of the many techniques to obtain a new convex function from the given functions is to calculate the product of these functions by imposing certain conditions on the functions. In general, the product of two or finite number of convex function needs not to be convex and, therefore, leads us to the study of product of these functions. In this paper, we reframe the idea of product of functions in the setting of generalized convex function to establish Hermite–Hadamard-type inequalities for these functions. We have analyzed different cases of double and triple integrals to derive some new results. The presented results can be viewed as the refinement and improvement of previously known results.


---

## Body

## 1. Introduction

Theory of convex functions has an essential role in different areas of mathematics, especially in optimization and modern analysis. Convex functions have many unique properties, for example, if a function is strictly convex, then it has a unique minimum on an open set. Even when the dimension of space is not finite, convex functions posses the similar properties and as a consequence, they are the examples of functionals in variation methods. A convex function via a random variable is bounded above by the expected value in the theory of probability. In fact, this is Jensen’s inequality, which can be used to reobtain many inequalities like the arithmetic-geometric mean inequality and Hölder’s inequality. One of the very fundamental results regarding convexity is the well-reputed Hermite–Hadamard inequality.(1)pξ1+ξ22≤1ξ2−ξ1∫ξ1ξ2pxdx≤pξ1+pξ22.Convexity has been generalized in many aspects, and the classical Hermite–Hadamard inequality is viewed by these generalizations. In [1], Toader extended the idea of convexity by giving the definition of an m-convex function and constructed few results including Hermite–Hadamard-type inequalities. Further development on m-convex and α,m-convex functions can be noticed in [1–9] and references therein. Further results on convex, m-convexn and α,m,h-convex functions can be seen in [10–21]. In [22], Pachpatte investigated the product of functions for developing Hermite–Hadamard-type inequalities by using the usual convexity. Recently, Noor et al. [23] introduced the idea of α,m,h-convex function and established few fundamental inequalities for the class of twice-differentiable functions. Since the α,m,h−convexity generalizes the concept of classical convexity, m-convexity, and α,m−convex functions, the results therein are generalized. Also, the convex functions and their described generalizations are characterized by the product of functions in an elegant way. Motivated by these generalizations and [1, 3, 22, 23], we utilize the idea of product of functions in the setting of α,m,h-convex function to establish Hermite–Hadamard-type inequalities for functions. We study different cases of double and triple integrals to derive some new results. This is the novel and innovative approach to characterize the convex function with the product of h−convex, m−convex, s,m−convex, and α,s−convex functions. Our results unify several classes of functions like s,m−Godunova–Levin, m−Godunova–Levin, and α,s−Godunova–Levin functions with mild conditions in an elegant way.


## 2. Preliminaries

This section is devoted to few well-known definitions fromthe literature. In [1], Toader gave the definition of m-convex function in the following manner.Definition 1.
A functionp:I⟶0,∞ is m-convex, if(2)pηx1+m1−ηx2≤tpx1+m1−ηpx2,for all x1,x2∈I, m∈0,1, and η∈0,1.
In [24], Mihesan extended the idea of m-convex functions by introducing the idea of α,m-convex function as follows.Definition 2.
A functionp:I⟶0,∞ is α,m-convex, if(3)pηx1+m1−ηx2≤ηαpx1+m1−ηαpx2,for all x1, x2∈I, η∈0,1, and α,m∈0,12.
In [23], Noor et al. generalized the idea of m-convexity and α,m-convexity in a more general way with the definition of α,m,h-convexity.Definition 3.
A functionp:I⟶0,∞ is α,m,h-convex, if(4)pηx1+m1−ηx2≤hηαpx1+mh1−ηαpx2,for all x1,x2∈I,η∈0,1,α,m∈0,12 and h:0,1⟶0,1.
In [22], Pachpatte used the idea of product of functions for convex functions to establish the following result.Theorem 1.
Letp and q be nonnegative and convex functions on ξ1,ξ2 and further assume that they are real valued. Then, the following two inequalities hold:(5)1ξ2−ξ1∫ξ1ξ2pxqxdx≤13Mξ1,ξ2+16Nξ1,ξ2,(6)2pξ1+ξ22qξ1+ξ22≤1ξ2−ξ1∫ξ1ξ2pxqxdx+16Mξ1,ξ2+13Nξ1,ξ2.


## 3. Main Results

This section contains the main results of our work involving product ofα,m,h−convex functions to obtain Hermite–Hadamard-type integral inequalities for two functions p and q. These inequalities have also been studied for double and triple integrals.Theorem 2.
Assume thatp and q are nonnegative and real-valued functions with pq∈Lξ1,ξ2, where ξ1,b∈I and ξ1<ξ2. Furthermore, assume that p and q are α,m,h-convex on ξ1,ξ2. Then, we have the following inequality:(7)1mξ2−ξ1∫ξ1mξ2pxqxdx≤hξ1,ξ1L+hξ2,ξ2M+hξ1,ξ2N.Here,(8)Hξ1,ξ1=pξ1qξ1,Hξ2,ξ2=pξ2qξ2,Hξ1,ξ2=pξ1qξ2+pξ2qξ1.
Also,L=∫01hηα2dη, M=∫01m2h1−ηα2dη and N=∫01mhηαh1−ηαdη.Proof.
Assume thatp and q are α,m,h-convex on ξ1,ξ2; then,(9)pηξ1+m1−ηξ2qηξ1+m1−ηξ2≤hηαpξ1+mh1−ηαpξ2hηαqξ1+mh1−ηαqξ2=hηα2pξ1qξ1+m2h1−ηα2pξ2qξ2+mhηα×h1−ηαpξ1qξ2+pξ2qξ1.
Now,(10)∫01pηξ1+m1−ηξ2qηξ1+m1−ηξ2dη≤∫01hηα2pξ1qξ1dη+∫01m2h1−ηα2pξ2qξ2dη+∫01mhηαh1−ηαpξ1qξ2+pξ2qξ1dη.
Thus,(11)∫01pηξ1+m1−ηξ2qηξ1+m1−ηξ2dη≤hξ1,ξ1L+hξ2,ξ2M+hξ1,ξ2N.
Now, substitutingηξ1+m1−ηξ2=x, we obtain(12)1mξ2−ξ1∫ξ1mξ2pxqxdx≤Hξ1,ξ1L+Hξ2,ξ2M+Hξ1,ξ2N.Corollary 1.
Ifhη=η, then(13)1mξ2−ξ1∫ξ1mξ2pxqxdx≤12α+1Hξ1,ξ1+2m2α22α+1α+1Hξ2,ξ2+mα2α+1α+1Hξ1,ξ2.Remark 1.
Corollary1, along with m=1 and α=1, gives inequality (5).Theorem 3.
For any twoα,m,h-convex functions p and q with pq∈Lξ1,ξ2, we have the following estimates:(14)2pξ1+mξ22qξ1+mξ22≤∫01pηξ1+n1−ηξ2qηξ1+m1−ηξ2dη+Hξ1,ξ1+n2Hξ2,ξ2S+12Hξ1,ξ2T,where(15)S=∫01hηαh1−ηαdη,T=∫01nhηα2+h1−ηα2dη.
AlsoHξ1,ξ1, Hξ2,ξ2, and Hξ1,ξ2 are as defined in the above theorem.Proof.
Asp and q are α,m,h-convex, we have(16)pξ1+mξ22qξ1+mξ22=pηξ1+m1−ηξ22+1−ηξ1+mηξ22×qηξ1+m1−ηξ22+1−ηξ1+mηξ22≤14pηξ1+m1−ηξ2+p1−ηξ1+mηξ2×qηξ1+m1−ηξ2+q1−ηξ1+mηξ2=14pηξ1+m1−ηξ2qηξ1+m1−ηξ2+p1−ηξ1+mηξ2q1−ηξ1+mηξ2+14pηξ1+m1−ηξ2q1−ηξ1+mηξ2+p1−ηξ1+mηξ2qηξ1+m1−ηξ2≤14pηξ1+m1−ηξ2qηξ1+m1−ηξ2+p1−ηξ1+mηξ2q1−ηξ1+mηξ2+14hηαpξ1+mh1−ηαpξ2×h1−ηαqξ1+mhηαqξ2≤14pηξ1+m1−ηξ2qηξ1+m1−ηξ2+p1−ηξ1+mηξ2q1−ηξ1+mηξ2+12hηαh1−ηαpξ1pξ2+m2pξ2qξ2+14mhηα2+h1−ηα2pξ1qξ2+pξ2qξ1.
On integrating,(17)pξ1+mξ22qξ1+mξ22≤14∫01pηξ1+m1−ηξ2qηξ1+m1−ηξ2+p1−ηξ1+mηξ2q1−ηξ1+mηξ2dη+12∫01hηαh1−ηαdηpξ1pξ2+m2pξ2qξ2+14∫01mhηα2+h1−ηα2dηpξ1qξ2+pξ2qξ1.
Thus, we obtain(18)pξ1+mξ22qξ1+mξ22≤12∫01pηξ1+m1−ηξ2qηξ1+m1−ηξ2dη+12Hξ1,ξ1+mHξ2,ξ2S+14Hξ1,ξ2T.
Now, substitutingx=ηξ1+n1−ηξ2, we obtain(19)2pξ1+mξ22qξ1+mξ22≤1ξ2−ξ1∫ξ1ξ2pxqxdη+hξ1,ξ1+nhξ2,ξ2S+12hξ1,ξ2T.Corollary 2.
Ifhη=η, then(20)2pξ1+mξ22qξ1+mξ22≤1ξ2−ξ1∫ξ1ξ2pxqxdη+αα+12α+1Hξ1,ξ1+mHξ2,ξ2+mα2+α+12α+12α+1+mα2+α+12α+12α+1Hξ1,ξ2.Remark 2.
Corollary2, along with m=1 and α=1, gives inequality (6).Theorem 4.
Assume thatp and q are α,m,h-convex functions satisfying all the conditions of the above theorem; then, the following inequality holds:(21)1ξ2−ξ12∫ξ1ξ2∫ξ1ξ2∫01pηx+m1−ηyqηx+m1−ηydηdydx≤L+M1ξ2−ξ1∫ξ1ξ2pxqxdx+121ξ2−ξ12NHξ1,ξ1+Hξ2,ξ2+Hξ1,ξ2,where(22)L=∫01hηα2dη,M=∫01m2h1−ηα2dη,N=∫01mhηαh1−ηαdη.Proof.
Using theα,m,h-convexity,(23)pηx+m1−ηyqηx+m1−ηy≤hηαpx+mh1−ηαpy×hηαqx+mh1−ηαqy=hηα2pxqx+m2h1−ηα×2pyqy+mhηαh1−ηαpxqy+pyqx.
Integrating on0,1, we obtain(24)∫01pηx+n1−ηyqηx+m1−ηydη≤pxqx∫01hηα2dη+pyqy∫01m2h1−ηα2dη+pxqy+pyqx∫01mhηαh1−ηαdη.
Thus,(25)∫01pηx+m1−ηyqηx+m1−ηydη≤pxqxL+pyqyM+pxqy+pyqxN.
Now, integrating on the rectangle0,1×0,1,(26)∫ξ1ξ2∫ξ1ξ2∫01pηx+m1−ηyqηx+m1−ηydηdydx≤Lξ2−ξ1∫ξ1ξ2pxqxdx+Mξ2−ξ1∫ξ1ξ2pyqydy+N∫ξ1xi2pxdx×∫ξ1ξ2qydy+∫ξ1ξ2pydy×∫ξ1ξ2qxdx.
Now, applying Hadamard’s inequality from right half to the above equation,(27)∫ξ1ξ2∫ξ1ξ2∫01pηx+m1−ηyqηx+m1−ηydηdydx≤L+Mξ2−ξ1∫ξ1ξ2pxqxdx+12Npξ1qξ1+pξ2qξ2+pξ1qξ2+pξ2qξ1=L+Mξ2−ξ1∫ξ1ξ2pxqxdx+12NHξ1,ξ1+Hξ2,ξ2+Hξ1,ξ2.
Thus, we obtain(28)1ξ2−ξ12∫ξ1ξ2∫ξ1ξ2∫01pηx+m1−ηyqηx+m1−ηydηdydx≤1ξ2−ξ1L+M∫ξ1ξ2pxqxdx+121ξ2−ξ12NHξ1,ξ1+Hξ2,ξ2+Hξ1,ξ2.Corollary 3.
Forhη=η, we have(29)1ξ2−ξ12∫ξ1ξ2∫ξ1ξ2∫01pηx+m1−ηyqηx+n1−ηydηdydx≤2α2m2+α+12α+1α+1ξ2−ξ1∫ξ1ξ2pxqxdx+mα22α+1α+1ξ2−ξ12Hξ1,ξ1+Hξ2,ξ2+Hξ1,ξ2.Remark 3.
Forα=1=m, we obtain equation (3) [22].Theorem 5.
Assume thatp and q are nonnegative real-valued functions such that they are α,m,h-convex and pq∈Lξ1,ξ2; then,(30)1ξ2−ξ1∫ξ1ξ2∫01pηx+m1−ηξ1+ξ22qηx+m1−ηξ1+ξ22dηdx≤Lξ2−ξ1∫ξ1ξ2pxqxdx+14MHξ1,ξ1+Hξ2,ξ2+Hξ1,ξ2+N2ξ2−ξ1Hξ1,ξ1+Hξ2,ξ2+Hξ1,ξ2,where L, M, and N are as in the above theorem.Proof.
Sincep and q are α,m,h-convex,(31)pηx+m1−ηξ1+ξ22qηx+m1−ηξ1+ξ22≤hηαpx+mh1−ηαpξ1+ξ22hηαqx+mh1−ηαqξ1+ξ22=hηα2pxqx+m2h1−ηα2pξ1+ξ22qξ1+ξ22+mhηαh1−ηαpxqξ1+ξ22+qxpξ1+ξ22.
This implies that(32)∫01pηx+n1−ηξ1+ξ22qηx+m1−ηξ1+ξ22dη≤pxqx∫01hηα2dη+pξ1+ξ22qξ1+ξ22∫01m2h1−ηα2dη+pxqξ1+ξ22+qxpξ1+ξ22∫01mhηαh1−ηαdη.
Integrating on0,1,(33)∫ξ1ξ2∫01pηx+n1−ηξ1+ξ22qηx+m1−ηξ1+ξ22dηdx≤L∫ξ1ξ2pxqxdx+Mξ2−ξ1pξ1+ξ22qξ1+ξ22+qξ1+ξ22∫ξ1ξ2pxdx+pξ1+ξ22∫ξ1ξ2qxdxN.
Now, using the right half of Hadamard’s inequality on the above equation,(34)∫ξ1ξ2∫01pηx+n1−ηξ1+ξ22qηx+m1−ηξ1+ξ22dvdx≤L∫ξ1ξ2pxqxdx+Mξ2−ξ1pξ1+pξ22qξ1+qξ22+2pξ1+pξ22qξ1+qξ22N.
Hence, we obtain(35)1ξ2−ξ1∫ξ1ξ2∫01pηx+m1−ηξ1+ξ22qηx+n1−ηξ1+ξ22dηdx≤Lξ2−ξ1∫ξ1ξ2pxqxdx+M4Hξ1,ξ1+Hξ2,ξ2+Hξ1,ξ2+N2ξ2−ξ1hξ1,ξ1+hξ2,ξ2+hξ1,ξ2.
This completes the proof.Corollary 4.
Ifhη=η, then(36)1ξ2−ξ1∫ξ1ξ2∫01pηx+n1−ηξ1+ξ22qηx+n1−ηξ1+ξ22dηdx≤1ξ2−ξ12α+1∫ξ1ξ2pxqxdx+α2m22α+12α+1hξ1,ξ1+hξ2,ξ2+hξ1,ξ2+mα2ξ2−ξ1α+12α+1hξ1,ξ1+hξ2,ξ2+hξ1,ξ2.Remark 4.
Forα=1=m, we obtain equation 4 of [22].


## 4. Conclusion

In this paper, we utilize the product of functions to develop the class of generalized convex functions using two given functions. We have studied this product forα,m,h−convex functions. Afterwards, we applied this to investigate Hermite–Hadamard inequalities of various types. We have analyzed that for the specific value of h, that is, to be identity function, these results coincide with the results for the product of α,m−convex functions. Moreover, the results are true for the said product in the sense of m-convexity with α=1. The comparison reflects that the obtained results improve and generalize the results for convex, m−convex, and α,m−convex functions in a peculiar way. For further interest of the readers in this direction, one may examine this product for invex, preinvex, m-preinvex, harmonically preinvex, and logarithmically preinvex functions. The idea is also interesting for fractional integrals and stochastic process for convex functions for new aspects in this regard [25].



---

*Source: 6630411-2021-01-27.xml*