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

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

For bounded non-negative operators and , Furuta showed

We will extend this as follows: implies

where is a harmonic mean of and . The idea of the proof comes from Jensen's inequality for an operator convex function by Hansen-Pedersen.

**1.**W. N. Anderson, G. E. Trapp, Shorted operators II, SIAM J. Appl. Math. 28(1975), 60-71.**2.**R. Bhatia,*Matrix Analysis*, Springer, 1996. MR**50:9417****3.**M. Fujii, T. Furuta, E. Kamei, Furuta's inequality and its application to Ando's theorem, Linear Alg. Appl. 179(1993), 161-169. MR**93j:47026****4.**T. Furuta, assures for with , Proc. Amer. Math. Soc. 101(1987), 85-88. MR**89b:47028****5.**T. Furuta, Two operator functions with monotone property, Proc. Amer. Math. Soc. 111(1991), 511-516. MR**91f:47023****6.**F. Hansen, G. K. Pedersen, Jensen's Inequality for Operators and Löwner's Theorem, Math. Ann. 258(1982), 229-241. MR**83g:47020****7.**F. Kubo, T. Ando, Means of positive linear operators, Math. Ann. 246(1980), 205-224. MR**84d:47028****8.**K. Tanahashi, Best possibility of Furuta inequality, Proc. Amer. Math. Soc. 124(1996), 141-146. MR**96d:47025****9.**M. Uchiyama, Some exponential operator inequalities, Math. Inequal. Appl. 2(1999), 469-471. MR**2000c:47036****10.**M. Uchiyama, Strong monotonicity of operator functions, Integr. Equ. Oper. Theory 37(1) (2000), 95-105. CMP**2000:14**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
47A63,
15A48

Retrieve articles in all journals with MSC (2000): 47A63, 15A48

Additional Information

**Mitsuru Uchiyama**

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

Email:
uchiyama@fukuoka-edu.ac.jp

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

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