An inequality for expectation of means of positive random variables

TL;DR

This paper establishes that the expectation of a mean of two positive random variables is bounded above by the mean of their expectations if and only if the mean is generated by a concave function, extending to operator means.

Contribution

It characterizes when the inequality for expectations of means holds, linking it to concavity and extending results to operator means in the Kubo-Ando framework.

Findings

The inequality holds iff the mean is generated by a concave function.

Extension of the inequality to operator means in the Kubo-Ando setting.

The harmonic mean case was previously proved by Rao and Prakasa Rao.

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.