New inequalities for the generalized Karcher mean

TL;DR

This paper introduces new inequalities for the generalized Karcher mean, exploring its properties, behavior under positive linear maps, and operator monotonicity, thereby advancing understanding of this mathematical concept.

Contribution

It provides novel inequalities and analyzes the behavior of the generalized Karcher mean, including its information monotonicity and operator monotone functions.

Findings

Established relations for the generalized Karcher mean

Analyzed the mean's behavior under positive linear maps

Revealed information monotonicity and operator monotonicity properties

Abstract

Recently, P\'{a}lfia introduced a generalized Karcher mean as a solution of an operator equation. In this article, we present several relations for this new mean. In particular, we investigate the behavior of this generalized mean when filtered through positive linear maps, thus its information monotonicity is revealed, and operator monotone function.