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

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.


Introduction
eory 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 arithmeticgeometric mean inequality and Hölder's inequality. One of the very fundamental results regarding convexity is the wellreputed Hermite-Hadamard inequality.
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][2][3][4][5][6][7][8][9] and references therein. Further results on convex, m-convexn and (α, m, h)-convex functions can be seen in [10][11][12][13][14][15][16][17][18][19][20][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. is is the novel and innovative approach to

Preliminaries
is section is devoted to few well-known definitions fromthe literature. In [1], Toader gave the definition of m-convex function in the following manner.
In [24], Mihesan extended the idea of m-convex functions by introducing the idea of (α, m)-convex function as follows.
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.
for all In [22], Pachpatte used the idea of product of functions for convex functions to establish the following result. Theorem 1. Let p and q be nonnegative and convex functions on [ξ 1 , ξ 2 ] and further assume that they are real valued. en, the following two inequalities hold:

Main Results
is section contains the main results of our work involving product of (α, m, h)−convex functions to obtain Hermite-Hadamard-type integral inequalities for two functions p and q. ese inequalities have also been studied for double and triple integrals.

Theorem 2. Assume that p and q are nonnegative and realvalued functions with pq
en, we have the following inequality: Here, Remark 1. Corollary 1, along with m � 1 and α � 1, gives inequality (5).
Proof. As p and q are (α, m, h)-convex, we have On integrating, us, we obtain

Theorem 4.
Assume that p and q are (α, m, h)-convex functions satisfying all the conditions of the above theorem; then, the following inequality holds: where Proof. Using the (α, m, h)-convexity, Integrating on [0, 1], we obtain Journal of Mathematics us,

(25)
Now, integrating on the rectangle [0, 1] × [0, 1], Now, applying Hadamard's inequality from right half to the above equation, us, we obtain 1 where L, M, and N are as in the above theorem. Proof. Since p and q are (α, m, h)-convex, is implies that Integrating on [0, 1],

Journal of Mathematics
Now, using the right half of Hadamard's inequality on the above equation, Hence, we obtain is completes the proof.

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 for (α, 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. e 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. e idea is also interesting for fractional integrals and stochastic process for convex functions for new aspects in this regard [25].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.