Year: 2025
Author: Xiaoyue Han, Run Xu
Journal of Nonlinear Modeling and Analysis, Vol. 7 (2025), Iss. 1 : pp. 135–177
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.
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.12150/jnma.2025.135
Journal of Nonlinear Modeling and Analysis, Vol. 7 (2025), Iss. 1 : pp. 135–177
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 43
Keywords: Discrete fractional calculus $h$-convex functions preinvex functions Hermite-Hadamard inequalities times scales.