Some New Discrete Hermite-Hadamard Inequalities and Their Generalizations (Journal of Nonlinear Modeling and Analysis, 2025)
This article mainly studies some new discrete Hermite-Hadamardinequalities for integer order and fractional order. For this purpose, the definitions of $h$-convexity and preinvexity for a real-valued function $f$ defined ona set of integers $\mathbb{Z}$ are introduced. Under these two new definitions, some newdiscrete Hermite-Hadamard inequalities for integer order related to the endpoints and the midpoint $\frac{a+b}{2}$ based on the substitution rules are proposed, andthey are generalized to fractional order forms. In addition, for the $h$-convexfunction on the time scale $\mathbb{Z},$ two new discrete Hermite-Hadamard inequalitiesfor integer order by dividing the time scale differently are obtained.
Keywords: Discrete fractional calculus; $h$-convex functions; preinvex functions; Hermite-Hadamard inequalities; times scales.