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

 

A decreasing operator function
associated with the Furuta inequality


Authors: Takayuki Furuta and Derming Wang
Journal: Proc. Amer. Math. Soc. 126 (1998), 2427-2432
MSC (1991): Primary 47A63
MathSciNet review: 1473667
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Abstract: Let $A\ge B\ge 0$ with $A>0$ and let $t\in[0,1]$ and $q\ge 0$. As a generalization of a result due to Furuta, it is shown that the operator function

\begin{displaymath}G_{p,q,t}(A,B,r,s)\!=\!A^{-r/2}\{A^{r/2} (A^{-t/2} B^p\!A^{-t/2})^s\! A^{r/2}\}^{(q-t+r)/[(p-t)s+r]}A^{-r/2} \end{displaymath}

is decreasing for $r\ge t$ and $s\ge 1$ if $p\ge\max\{q,t\}$. Moreover, if $1\ge p>t$ and $q\ge t$, then $G_{p,q,t}(A,B,r,s)$ is decreasing for $r\ge 0$ and $s\ge \frac{q-t}{p-t}$. The latter result is an extension of an earlier result of Furuta.


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Additional Information

Takayuki Furuta
Affiliation: Department of Applied Mathematics, Faculty of Science, Science University of Tokyo, Kagurazaka, Shinjuku 162-8601, Tokyo, Japan
Email: furuta@rs.kagu.sut.ac.jp

Derming Wang
Affiliation: Department of Mathematics, California State University, Long Beach, Long Beach, California 90840-1001

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04632-2
PII: S 0002-9939(98)04632-2
Keywords: L\"owner-Heinz inequality, Furuta inequality
Received by editor(s): January 23, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society