A Generalization of Hermite-Hadamard's Inequality

TL;DR

This paper introduces a generalized Hermite-Hadamard inequality that unifies the midpoint and trapezoid inequalities, extending classical results and deriving new Ostrowski-type inequalities for convex functions.

Contribution

It presents a new inequality that generalizes Hermite-Hadamard by combining composite trapezoid and midpoint formulas, with classical inequality as a special case.

Findings

A new Hermite-Hadamard-like inequality is proved.

Classical Hermite-Hadamard inequality is a special case.

Ostrowski's type inequalities for convex functions are derived.

Abstract

In literature the Hermite-Hadamard inequality was eligible for many reasons, one of the most surprising and interesting that the Hermite-Hadamard inequality combine the midpoint and trapezoid formulae in an inequality. In this work, a Hermite-Hadamard like inequality that combines the composite trapezoid and composite midpoint formulae is proved. So that, the classical Hermite-Hadamard inequality becomes a special case of the presented result. Some Ostrowski's type inequalities for convex functions are proved as well.