Operator inequalities among arithmetic mean, geometric mean and harmonic mean

TL;DR

This paper establishes new bounds for the weighted geometric mean of two invertible positive operators using the weighted arithmetic and harmonic means, enhancing the understanding of operator inequalities.

Contribution

It provides novel upper and lower bounds for the weighted geometric mean in the context of operator inequalities, extending existing mathematical frameworks.

Findings

Derived an upper bound for the weighted geometric mean.

Established a lower bound for the weighted geometric mean.

Applicable to invertible positive operators.

Abstract

We give an upper bound for the weighted geometric mean using the weighted arithmetic mean and the weighted harmonic mean. We also give a lower bound for the weighted geometric mean. These inequalities are proven for two invertible positive operators.