Some New Discrete Hermite-Hadamard Inequalities and Their Generalizations

Abstract

This article mainly studies some new discrete Hermite-Hadamard inequalities for integer order and fractional order. For this purpose, the definitions of $h$-convexity and preinvexity for a real-valued function $f$ defined on a set of integers $\mathbb{Z}$ are introduced. Under these two new definitions, some new discrete Hermite-Hadamard inequalities for integer order related to the endpoints and the midpoint $\frac{a+b}{2}$ based on the substitution rules are proposed, and they are generalized to fractional order forms. In addition, for the $h$-convex function on the time scale $\mathbb{Z},$ two new discrete Hermite-Hadamard inequalities for integer order by dividing the time scale differently are obtained.