A Note on the Weighted Harmonic-Geometric-Arithmetic Means Inequalities

TL;DR

This paper derives sharp bounds for weighted harmonic-geometric-arithmetic means when two terms are known, with applications to matrix trace bounds and polynomial coefficients.

Contribution

It introduces new sharp bounds for means inequalities and applies them to matrix analysis and polynomial coefficient inequalities.

Findings

Derived explicit bounds for matrix trace of inverse.

Established inequalities for polynomial coefficients with positive roots.

Provided sharp bounds for weighted means with partial information.

Abstract

In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of a symmetric positive definite matrix and an inequality related to the coefficients of polynomials with positive roots.