Jensen's operator inequality

for functions of several variables

Authors:
Huzihiro Araki and Frank Hansen

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2075-2084

MSC (1991):
Primary 47A63; Secondary 47A80, 47Bxx

DOI:
https://doi.org/10.1090/S0002-9939-00-05371-5

Published electronically:
February 16, 2000

MathSciNet review:
1670414

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The operator convex functions of several variables are characterized in terms of a non-commutative generalization of Jensen's inequality, extending previous results of F. Hansen and G.K. Pedersen for functions of one variable and of F. Hansen for functions of two variables.

**1.**T. Ando. Concavity of certain maps of positive definite matrices and applications to adamard products.*Linear Algebra Appl.*, 26:203-241, 1979. MR**80f:15023****2.**MR**94h:15008**J.S. Aujla. Matrix convexity of functions of two variables.*Linear Algebra and Its Applications*, 194:149-160, 1993. MR**94h:15008****3.**P.R. Halmos.*A Hilbert space problem book*. Van Nostrand, 1967. MR**34:8178****4.**F. Hansen. An operator inequality.*Math. Ann.*, 246:249-250, 1980. MR**82a:46065****5.**F. Hansen. Jensen's operator inequality for functions of two variables.*Proc. Amer. Math. Soc.*, 125:2093-2102, 1997. MR**97i:47027****6.**F. Hansen. Operator convex functions of several variables.*Publ. Res. Inst. Math. Sci.*, 33:443-463, 1997. MR**99b:47022b**; MR**99b:47022a****7.**F. Hansen and G.K. Pedersen. Jensen's inequality for operators and öwner's theorem.*Math. Ann.*, 258:229-241, 1982. MR**83g:47020****8.**A. Korányi. On some classes of analytic functions of several variables.*Trans Amer. Math. Soc.*, 101:520-554, 1961. MR**25:226****9.**G.K. Pedersen. Extreme -tuples of elements in -algebras.*Bull. London Math. Soc.*, 19:264-270, 1987. MR**88m:46070**

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Additional Information

**Huzihiro Araki**

Affiliation:
Department of Mathematics, Faculty of Science and Technology, The Science University of Tokyo, Noda, Chiba-ken 278-8510, Japan

Email:
araki@ma.noda.sut.ac.jp

**Frank Hansen**

Affiliation:
Institute of Economics, University of Copenhagen, Studiestraede 6, 1455 Copenhagen K, Denmark

Email:
frank.hansen@econ.ku.dk

DOI:
https://doi.org/10.1090/S0002-9939-00-05371-5

Received by editor(s):
August 30, 1998

Published electronically:
February 16, 2000

Communicated by:
David R. Larson

Article copyright:
© Copyright 2000
American Mathematical Society