Non-commutative A-G mean inequality

Abstract

In this paper we consider non-commutative analogue for the arithmeticgeometric mean inequality $$a^r{b}^{1-r}+(r-1)b\ge ra$$ 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.

10 pages