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Proceedings of the American Mathematical Society

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Non-commutative arithmetic-geometric mean inequality

Author: Tomohiro Hayashi
Journal: Proc. Amer. Math. Soc. 137 (2009), 3399-3406
MSC (2000): Primary 47A63, 47A64
Published electronically: May 6, 2009
MathSciNet review: 2515409
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Abstract: In this paper we consider a non-commutative analogue of the arithmetic-geometric mean inequality

$\displaystyle a^{r}b^{1-r}+(r-1)b\geq ra$

for two positive numbers $ a,b$ and for $ r> 1$. We show that under certain assumptions the non-commutative analogue of $ a^{r}b^{1-r}$ which satisfies this inequality is unique and equal to the $ 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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Additional Information

Tomohiro Hayashi
Affiliation: Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555, Japan

Keywords: Operator inequality, operator mean, geometric mean
Received by editor(s): May 1, 2008
Received by editor(s) in revised form: February 12, 2009
Published electronically: May 6, 2009
Communicated by: Marius Junge
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.