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arXiv · 1608.08671

An inequality for expectation of means of positive random variables

Paolo Gibilisco·Frank Hansen
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Abstract

Suppose that $X,Y$ are positive random variable and $m$ a numerical (commutative) mean. We prove that the inequality ${\rm E} (m(X,Y)) \leq m({\rm E} (X), {\rm E} (Y))$ 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.

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BibTeXRIS

Paolo Gibilisco, Frank Hansen. 2016-08-30. An inequality for expectation of means of positive random variables. https://doi.org/10.1215/20088752-3750087

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