TY - JOUR
T1 - Some New Discrete Hermite-Hadamard Inequalities and Their Generalizations
AU - Han , Xiaoyue
AU - Xu , Run
JO - Journal of Nonlinear Modeling and Analysis
VL - 1
SP - 135
EP - 177
PY - 2025
DA - 2025/02
SN - 7
DO - http://doi.org/10.12150/jnma.2025.135
UR - https://global-sci.org/intro/article_detail/jnma/23837.html
KW - Discrete fractional calculus, $h$-convex functions, preinvex functions, Hermite-Hadamard inequalities, times scales.
AB - <p style="text-align: justify;">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.</p>