Hermite-Hadamard (HH) integral inequality for m,q-preinvex functions

In recent years, convexity theory has experienced rapid development. Many researchers have expanded it. A notable generalization of the convex function is the Inox function introduced and studied by Hanson. This has greatly expanded the role of inox in optimization. Ben and Mond introduced a class of functions called generalizations of pre-Inox functions. Regarding change inequalities and related problems in recent years, Pite and Postloche introduced the concept of pseudo pre-invex, and applied it in theoretical mechanics and nonlinear optimization. Later, Pite and Antchak introduced this concept of invex, and applied it to vector optimization. This shows that pre-invexity plays an important role in the development of various fields of pure and applied sciences. In this article, we first introduce the concept of  -pre-inox and then state and prove the Hermit-Hadamard theorem for it, and we will state that the previously obtained inequalities are a direct result of our main theorem. In this article, we first introduce the concept of  invex sets and  pre-invex functions. Then state and prove the Hermit-Hadamard theorem for it, and state that the previous inequalities are a direct result of our main theorem. Quantum calculus is known as the study of differential and integral calculus without restrictions. Euler (1783-1707) was the first to study quantum calculus. He introduced q in Newton's infinite series compositions. In the early 20th century, the study of quantum calculus was started by Jackson. In quantum computing, we obtain mathematical equivalents of q objects that can be recovered as . Note that quantum calculus is a subset of time scale calculus. The time scale of calculus provides a unified framework for the study of dynamical equations in both discrete and continuous domains. That in quantum calculus, we deal with a specific time scale called the q time scale. Quantum calculus is a bridge between mathematics and physics. Due to the significant applications of quantum computing in mathematics and physics, this issue has been the focus of many researchers. As a result, quantum calculus has emerged as a fascinating field. In recent years, convexity theory has experienced rapid development. Many researchers have expanded it. A notable generalization of the convex function is the inex function, which was introduced and studied by Hanson [1]. This has greatly expanded the role of inox in optimization. Ben and Mond [2] introduced a class of functions called generalizations of pre-invex functions.This fundamental result of Hermit and Hadamard (HH) has obtained by many mathematicians, and as a result, this inequality has been extended by Noor in various ways using new ideas and obtained the Hermit-Hadamard inequality for pre-invex functions. Regarding variational inequalities and related problems in recent years, Pite and Postloche [3],[4]  and [5] introduced the concept of pseudo pre-invex, and applied it in theoretical mechanics and nonlinear optimization. Later, Pite and Antchak [6] introduced this concept of inexity, and applied it in vector optimization. This shows that pre-invex plays an important role in the development of various fields of pure and applied sciences. For more details on the quantum calculus see references [7].MethodNo method applicable. In This article, we improve the Hermite-Hadamard (HH) integral inequality for  - preinvex functions.Assuming , Theorem 3.1 of the article [8], that is, relation (3) is obtained. Assuming , the main theorem of the article [9] and  and , Hermit Hadamard's main inequality is obtained.