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Mean theoretic approach to the grand Furuta inequality
Authors:Masatoshi Fujii   Eizaburo Kamei
Affiliation:Department of Mathematics, Osaka Kyoiku University, Asahigaoka, Kashiwara, Osaka 582, Japan ; Momodani Senior Highschool, Ikuno, Osaka 544, Japan
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.

Keywords:Positive operators, L"  {o}wner-Heinz inequality, Furuta inequality, Ando-Hiai inequality, grand Furuta inequality
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