Some integral inequalities of Hermite-Hadamard type for functions whose derivatives of $n$-th order are $(\alpha,m)$-convex

TL;DR

This paper establishes new Hermite-Hadamard type integral inequalities for functions with derivatives of order n that are $(eta,m)$-convex, extending known results and applying them to inequalities involving special means of positive numbers.

Contribution

It introduces novel integral inequalities for higher-order derivatives with $(eta,m)$-convexity, generalizing existing Hermite-Hadamard inequalities and deriving applications to special means.

Findings

New inequalities for derivatives of order n with $(eta,m)$-convexity

Generalizations of Hermite-Hadamard inequalities

Applications to inequalities involving special means

Abstract

In the paper, the authors find some new integral inequalities of Hermite-Hadamard type for functions whose derivatives of the $n$-th order are $(\alpha,m)$-convex and deduce some known results. As applications of the newly-established results, the authors also derive some inequalities involving special means of two positive real numbers.