Hermite-Hadamard inequalities for (p,a,b)-convex functions

TL;DR

This paper introduces Hermite-Hadamard inequalities tailored for a new class of functions called (p,a,b)-convex functions, providing tighter bounds and extending results to fractional integrals.

Contribution

The paper establishes novel Hermite-Hadamard inequalities for (p,a,b)-convex functions, enhancing classical bounds and including fractional integral inequalities.

Findings

Derived tighter Hermite-Hadamard inequalities for (p,a,b)-convex functions.

Extended inequalities to fractional integrals involving these functions.

Demonstrated the applicability of inequalities to broader function classes.

Abstract

A function $f:[a,b] \rightarrow \mathbb{R}$ is called $(p,a,b)$-convex if $f$ is $p$ times continuously differentiable, $f^{(p)}$ is convex and increasing, and $f^{(k)}(a)=0$ for all $k=1,\ldots,p$ where $f^{(j)}$ is the $j$th derivative of $f$. In this note we prove Hermite-Hadamard inequalities for $(p,a,b)$-convex functions that are significantly tighter than the classical Hermite-Hadamard inequality. We also prove inequalities for fractional integrals that involve $(p,a,b)$-convex functions.