The Hermite-Hadamard inequality revisited: Some new proofs and applications

TL;DR

This paper presents new proofs of the Hermite-Hadamard inequality and explores various applications, including inequalities for functions with specific derivative properties and estimates for moments, along with a reverse Hardy inequality for convex functions.

Contribution

It offers novel proofs of the Hermite-Hadamard inequality and introduces new applications and bounds related to convex and inflection point functions.

Findings

New proofs of Hermite-Hadamard inequality

Hadamard-type inequalities for derivative-inflexion functions

Estimates for moments and reverse Hardy inequality

Abstract

New proofs of the classical Hermite-Hadamard inequality are presented and several applications are given, including Hadamard-type inequalities for the functions, whose derivatives have inflection points or whose derivatives are convex. Morever, some estimates from below and above for the first moments of functions $% f:[a,b]\rightarrow %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $ about the center point $c=(a+b)/2$ are obtained and the reverse Hardy inequality for convex functions $f:[0,\infty )\rightarrow (0,\infty )$ is established.