Mean theoretic approach to the grand Furuta inequality

Fujii, Masatoshi; Kamei, Eizaburo

doi:10.1090/S0002-9939-96-03342-4

Mean theoretic approach to the grand Furuta inequality
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by Masatoshi Fujii and Eizaburo Kamei

Proc. Amer. Math. Soc. 124 (1996), 2751-2756

DOI: https://doi.org/10.1090/S0002-9939-96-03342-4

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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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