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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Mean theoretic approach
to the grand Furuta inequality


Authors: Masatoshi Fujii and Eizaburo Kamei
Journal: Proc. Amer. Math. Soc. 124 (1996), 2751-2756
MSC (1991): Primary 47A63, 47B15
MathSciNet review: 1327013
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Abstract: Very recently, Furuta obtained the grand Furuta inequality which is a parameteric formula interpolating the Furuta inequality and the Ando-Hiai inequality as follows : If $A \ge B \ge 0$ and $A$ is invertible, then for each $t \in [0,1]$,

\begin{equation*}F_{p,t}(A,B,r,s) = A^{-r/2}\{A^{r/2}(A^{-t/2}B^{p}A^{-t/2})^{s}A ^{r/2}\}^{\frac {1-t+r}{(p-t)s+r}}A^{-r/2} \end{equation*}

is a decreasing function of both $r$ and $s$ for all $r \ge t, ~p \ge 1$ and $s \ge 1$. In this note, we employ a mean theoretic approach to the grand Furuta inequality. Consequently we propose a basic inequality, by which we present a simple proof of the grand Furuta inequality.


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Additional Information

Masatoshi Fujii
Affiliation: Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582, Japan
Email: mfujii@cc.osaka-kyoiku.ac.jp

Eizaburo Kamei
Affiliation: Momodani Senior Highschool, Ikuno, Osaka 544, Japan

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03342-4
PII: S 0002-9939(96)03342-4
Keywords: Positive operators, L\"{o}wner-Heinz inequality, Furuta inequality, Ando-Hiai inequality, grand Furuta inequality
Received by editor(s): November 28, 1994
Received by editor(s) in revised form: March 6, 1995
Dedicated: Dedicated to Professor Tsuyoshi Ando, the originator of the theory of operator means, on his retirement from Hokkaido University
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society