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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An operator inequality related to Jensen’s inequality
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by Mitsuru Uchiyama PDF
Proc. Amer. Math. Soc. 129 (2001), 3339-3344 Request permission

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
  • Received by editor(s): March 21, 2000
  • Published electronically: April 9, 2001
  • Communicated by: Joseph A. Ball
  • © Copyright 2001 American Mathematical Society
  • 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
  • MathSciNet review: 1845011