Suppose that X, Y are positive random variables and m is a numerical (commutative) mean. We prove that the inequality E(m(X, Y )) leq 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.
Gibilisco, P., Hansen, F. (2017). An inequality for expectation of means of positive random variables. ANNALS OF FUNCTIONAL ANALYSIS, 8(1), 142-151 [10.1215/20088752-3750087].
An inequality for expectation of means of positive random variables
Gibilisco P.
;
2017-01-01
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 )) leq 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.File in questo prodotto:
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