Annals of Functional Analysis

An Inequality for expectation of means of positive random variables

Paolo Gibilisco and Frank Hansen

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Abstract

Suppose that X, Y are positive random variables and m is a numerical (commutative) mean. We prove that the inequality E(m(X,Y))m(E(X),E(Y)) holds if and only if the mean is generated by a concave function. With due changes we also prove that the same inequality holds for all operator means in the Kubo–Ando setting. The case of the harmonic mean was proved by C. R. Rao and B. L. S. Prakasa Rao.

Article information

Source
Ann. Funct. Anal., Volume 8, Number 1 (2017), 142-151.

Dates
Received: 9 June 2016
Accepted: 1 August 2016
First available in Project Euclid: 12 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1478919628

Digital Object Identifier
doi:10.1215/20088752-3750087

Mathematical Reviews number (MathSciNet)
MR3572337

Zentralblatt MATH identifier
1354.26053

Subjects
Primary: 26E60: Means [See also 47A64]
Secondary: 47A64: Operator means, shorted operators, etc. 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52)

Keywords
numerical means operator means concavity random matrices

Citation

Gibilisco, Paolo; Hansen, Frank. An Inequality for expectation of means of positive random variables. Ann. Funct. Anal. 8 (2017), no. 1, 142--151. doi:10.1215/20088752-3750087. https://projecteuclid.org/euclid.afa/1478919628


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