Generalized Fej\'er-Hermite-Hadamard type via generalized $(h-m)$-convexity on fractal sets and applications

TL;DR

This paper introduces a new class of convex functions called generalized (h-m)-convexity on fractal sets, extending classical inequalities like Hermite-Hadamard and Fejér-Hermite-Hadamard, with applications to random variables and numerical integration.

Contribution

It defines generalized (h-m)-convexity on fractal sets and extends key inequalities to this new class, including applications to various fields.

Findings

Established properties of generalized (h-m)-convexity.

Derived generalized Hermite-Hadamard and Fejér-Hermite-Hadamard inequalities.

Applied results to random variables and numerical integration.

Abstract

In this article, we define a new class of convexity called generalized $(h-m)$-convexity, which generalizes $h$-convexity and $m$-convexity on fractal sets $\mathbb{R}^{\alpha}$ $(0<\alpha\leq 1)$. Some properties of this new class are discussed. Using local fractional integrals and generalized $(h-m)$-convexity, we generalized Hermite-Hadamard (H-H) and Fej\'er-Hermite-Hadamard (Fej\'er-H-H) types inequalities. We also obtained a new result of the Fej\'er-H-H type for the function whose derivative in absolute value is the generalized $(h-m)$-convexity on fractal sets. Some applications to random variables and numerical integrations are studied.