# Non-commutative A-G mean inequality

@article{Hayashi2008NoncommutativeAM, title={Non-commutative A-G mean inequality}, author={T. Hayashi}, journal={arXiv: Functional Analysis}, year={2008} }

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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