(untitled) (Journal of Nonlinear Modeling and Analysis, 2025)
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 Z are introduced. Under these two new definitions, some new discrete Hermite-Hadamard inequalities for integer order related to the endpoints and the midpoint a+b2 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 Z, two new discrete Hermite-Hadamard inequalities for integer order by dividing the time scale differently are obtained.