An Inequality for expectation of means of positive random variables

Abstract

Suppose that $X$, $Y$ are positive random variables and $m$ is a numerical (commutative) mean. We prove that the inequality $\mathrm{E}\left(m\right(X,Y\left)\right)\le m\left(\mathrm{E}\right(X),\mathrm{E}(Y\left)\right)$ 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.