Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



An operator inequality related to Jensen's inequality

Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 129 (2001), 3339-3344
MSC (2000): Primary 47A63, 15A48
Published electronically: April 9, 2001
MathSciNet review: 1845011
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For bounded non-negative operators $A$ and $B$, Furuta showed

\begin{displaymath}0\leq A \leq B {\rm implies } A^{\frac{r}{2}}B^sA^{\frac{r}{2... ... A^{\frac{r}{2}})^{\frac{s+r}{t+r}} (0\leq r, 0\leq s \leq t).\end{displaymath}

We will extend this as follows: $0\leq A\leq B \underset{\lambda}{!}C $ $(0<\lambda <1)$ implies

\begin{displaymath}A^{\frac{r}{2}}(\lambda B^s+ (1-\lambda)C^s)A^{\frac{r}{2}} \... ...bda B^t+ (1- \lambda)C^t) A^{\frac{r}{2}}\}^{\frac{s+r}{t+r}} ,\end{displaymath}

where $B \underset{\lambda}{!}C$ is a harmonic mean of $B$ and $C$. The idea of the proof comes from Jensen's inequality for an operator convex function by Hansen-Pedersen.

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Additional Information

Mitsuru Uchiyama
Affiliation: Department of Mathematics, Fukuoka University of Education, Munakata, Fukuoka, 811-4192, Japan

Keywords: Order of selfadjoint operators, Jensen inequality, Furuta inequality
Received by editor(s): March 21, 2000
Published electronically: April 9, 2001
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society