Weak log-majorization and inequalities of power means

TL;DR

This paper explores weak log-majorization and inequalities related to various power means, including Lim-Pálfia's and Rényi means, establishing new relationships and bounds in the context of non-commutative quasi-arithmetic means.

Contribution

It introduces novel weak log-majorization results and inequalities for non-commutative power means, extending classical mean inequalities to new operator settings.

Findings

Lim-Pálfia's power mean of order t in [-1,0) is weakly log-majorized by the log-Euclidean mean.

The paper establishes the Ando-Hiai inequality for these means.

It provides bounds for Rényi power means in terms of quasi-arithmetic means.

Abstract

As non-commutative versions of the quasi-arithmetic mean, we consider the Lim-P\'{a}lfia's power mean, R\'{e}nyi right mean and R\'{e}nyi power means. We prove that the Lim-P\'{a}lfia's power mean of order $t \in [-1,0)$ is weakly log-majorized by the log-Euclidean mean and fulfills the Ando-Hiai inequality. We establish the log-majorization relationship between the R\'{e}nyi relative entropy and the product of square roots of given variables. Furthermore, we show the norm inequalities among power means and provide the boundedness of R\'{e}nyi power mean in terms of the quasi-arithmetic mean.