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

DOI:
https://doi.org/10.1090/S0002-9939-01-06130-5

Published electronically:
April 9, 2001

MathSciNet review:
1845011

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Abstract | References | Similar Articles | Additional Information

Abstract: For bounded non-negative operators $A$ and $B$, Furuta showed \[ 0\leq A \leq B \ \textrm {implies } \ A^{\frac {r}{2}}B^sA^{\frac {r}{2}} \leq (A^{\frac {r}{2}}B^t A^{\frac {r}{2}})^{\frac {s+r}{t+r}} \ \ (0\leq r, \ 0\leq s \leq t).\] We will extend this as follows: $0\leq A\leq B \underset {\lambda }{!}C$ $(0<\lambda <1)$ implies \[ A^{\frac {r}{2}}(\lambda B^s+ (1-\lambda )C^s)A^{\frac {r}{2}} \leq \{A^{\frac {r}{2}} (\lambda B^t+ (1- \lambda )C^t) A^{\frac {r}{2}}\}^{\frac {s+r}{t+r}} ,\] 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

MR Author ID:
198919

Email:
uchiyama@fukuoka-edu.ac.jp

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