# Generalized Convex Functions and Their Applications

## TL;DR

This paper introduces generalized convex functions, explores their properties, and derives new inequalities related to Hermite-Hadamard type, with applications to special means.

## Contribution

It extends the concept of convexity to generalized and s-convex functions, establishing new inequalities and properties in this broader context.

## Key findings

- Established properties of generalized convex functions.
- Derived new Hermite-Hadamard type inequalities.
- Applied inequalities to special means.

## Abstract

This study focuses on convex functions and their generalized. Thus, we start this study by giving the definition of convex functions and some of their properties and discussing a simple geometric property. Then we generalize E-convex functions and establish some their properties. Moreover, we give generalized $ s $-convex functions in the second sense and present some new inequalities of generalized Hermite-Hadamard type for the class of functions whose second local fractional derivatives of order $ \alpha $ in absolute value at certain powers are generalized $ s $-convex functions in the second sense. At the end, some examples that these inequalities are able to be applied to some special means are showed.