Non-commutative arithmetic-geometric mean inequality
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- by Tomohiro Hayashi PDF
- Proc. Amer. Math. Soc. 137 (2009), 3399-3406 Request permission
Abstract:
In this paper we consider a non-commutative analogue of the arithmetic-geometric mean inequality \[ a^{r}b^{1-r}+(r-1)b\geq ra\] 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.References
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Additional Information
- Tomohiro Hayashi
- Affiliation: Nagoya Institute of Technology, Gokiso-cho, Showa-ku, Nagoya, Aichi, 466-8555, Japan
- Email: hayashi.tomohiro@nitech.ac.jp
- 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
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 3399-3406
- MSC (2000): Primary 47A63, 47A64
- DOI: https://doi.org/10.1090/S0002-9939-09-09911-0
- MathSciNet review: 2515409