On new types of convex functions and their properties

TL;DR

This paper introduces new convex functions and lemmas, explores their relation to Caputo derivatives, and revises the Hermite-Hadamard inequality based on these new definitions.

Contribution

It presents novel convex functions and lemmas, linking them to fractional derivatives and extending classical inequalities.

Findings

New convex functions for specific parameter ranges

Relevance established between lemmas and Caputo derivatives

Revised Hermite-Hadamard inequality based on new definitions

Abstract

In this study, Firstly, we will write two new convex functions for $-1<n-\alpha \leq 1\ $and two new lemmas. Then we will find the relevance of the two new lemmas to Caputo-left-sided derivatives under additional conditions and draw conclusions about them. Secondly,We will give examples that support it. In what follows, we will rewrite the Hermite Hadamard integral inequality based on this new definitions. Furthermore, we will write an elementary inequality valid for value $-1<n-\alpha \leq 1.$