On the power means and Lawson-Lim means for positive invertible operators

TL;DR

This paper explores reverse inequalities involving power means and Karcher mean for positive invertible operators, generalizing existing inequalities and comparing different operator means.

Contribution

It introduces new reverse inequalities for power and Karcher means, extending previous results to higher powers and providing comparisons between different operator means.

Findings

Reverse inequalities for power means and Karcher mean established.

Generalization of Lawson-Lim inequality to higher powers.

Comparison between Karcher and Lawson-Lim means for higher powers.

Abstract

This note aims to present some reverse inequalities about the power means and Karcher mean via the Kantorovich constant and some of these have been generalized to higher power. Also, we generalize the reverse weighted arithmetic-geometric mean inequality of n positive invertible operators due to Lawson and Lim. In addition, we make comparisons between the Karcher mean and Lawson-Lim geometric mean for higher power.