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| Some New Discrete Hermite-Hadamard Inequalities and Their Generalizations |
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| KeyWord:Discrete fractional calculus, $h$-convex functions, preinvex functions, Hermite-Hadamard inequalities, times scales |
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| 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. |
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