New Hermite-Hadamard Type Inequalities for Twice Differentiable Composite $(h-s)_2$-Convex Functions

TL;DR

This paper extends Hermite-Hadamard inequalities to twice differentiable composite functions with specific convexity properties, addressing open problems and applying results to special means of real numbers.

Contribution

It provides new Hermite-Hadamard inequalities for composite functions with second derivatives that are ((h-s)_2, I)-convex, solving open problems from prior research.

Findings

Derived new inequalities for twice differentiable composite functions.

Applied inequalities to special means of real numbers.

Addressed open problems regarding composition of convex functions.

Abstract

In a recent paper [9], Ozdemir, Tunc and Akdemir defined two new classes of convex functions with which they proved some Hermite-Hadamard type inequalities. As an Open problem, they asked for conditions under which the composition of two functions belong to their newly defined class of convex functions and if Hermite-Hadamard type inequalities can be obtained. In this paper, we respond to the Open problems and prove some new Hermite-Hadamard inequalities for twice differentiable composition whose second derivative is $((h-s)_{2}, I)$-convex. Our results are applied to some special means of real numbers.