Non-commutative A-G mean inequality

TL;DR

This paper explores a non-commutative version of the arithmetic-geometric mean inequality, establishing uniqueness of the mean under certain conditions and providing a new characterization of the geometric mean.

Contribution

It introduces a non-commutative analogue of the inequality, proves its uniqueness as the r-mean, and extends the understanding of geometric mean characterization.

Findings

Uniqueness of the non-commutative r-mean under certain assumptions

Extension of the inequality to the case 0<r<1

New characterization of the geometric mean

Abstract

In this paper we consider non-commutative analogue for the arithmeticgeometric mean inequality $$a^{r}b^{1-r}+(r-1)b\geq ra$$ for two positive numbers $a,b$ and $r> 1$. We show that under some assumptions the non-commutative analogue for $a^{r}b^{1-r}$ which satisfies this inequality is unique and equal to $r$-mean. The case $0<r<1$ is also considered. In particular, we give a new characterization of the geometric mean.