An inequality for expectation of means of positive random variables

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Suppose that X,Y are positive random variable and m 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.