Annals of Functional Analysis

An Inequality for expectation of means of positive random variables

Paolo Gibilisco and Frank Hansen

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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

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

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

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

numerical means operator means concavity random matrices


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.

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