# 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.

## Key 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.