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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
DOI: https://doi.org/10.1090/S0002-9939-01-06130-5
Published electronically: April 9, 2001
MathSciNet review: 1845011
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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