Some refinements of Hermite-Hadamard inequality and an open problem

TL;DR

This paper refines the Hermite-Hadamard inequality using linear combinations of endpoints, explores optimal constants related to Simpson's rule, and provides solutions for specific convex functions.

Contribution

It introduces a new refinement of the Hermite-Hadamard inequality and addresses the problem of best constants for a broad class of convex functions.

Findings

Refined Hermite-Hadamard inequality as a linear combination of endpoints

Solved the best constant problem for certain convex functions

Provided supplementary results related to numerical integration

Abstract

We presented here a refinement of Hermite-Hadamard inequality as a linear combination of its end-points. The problem of best possible constants is closely connected with well known Simpson's rule in numerical integration. It is solved here for a wide class of convex functions, but not in general. Some supplementary results are also given.