Means refinements via convexity

TL;DR

This paper develops convex inequalities to precisely compare various means, refining and unifying classical inequalities involving arithmetic, geometric, harmonic, and Heinz means for numbers and operators.

Contribution

It introduces a convex inequality framework that refines and unifies existing inequalities among different means, providing exact difference limits between convex functions and their secants.

Findings

Derived a convex inequality expressing the difference between a convex function and its secant as a limit.

Unified and refined inequalities involving arithmetic, geometric, harmonic, and Heinz means.

Applicable to both numbers and operators.

Abstract

The main goal of this article is to find the exact difference between a convex function and its secant, as a limit of positive quantities. This idea will be expressed as a convex inequality that leads to refinements and reversals of well established inequalities treating different means. The significance of these inequalities is to write one inequality that brings together and refine almost all known inequalities treating the arithmetic, geometric, harmonic and Heinz means, for numbers and operators.