Non-commutative A-G mean inequality

Tomohiro Hayashi

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arXiv · 0805.0071

Non-commutative A-G mean inequality

Tomohiro Hayashi

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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.

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Tomohiro Hayashi. 2008-05-01. Non-commutative A-G mean inequality. https://arxiv.org/abs/0805.0071

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