An inequality for expectation of means of positive random variables

Abstract

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

References in corpus (1)

• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities

Cited by in corpus (1)

• A unified approach to Local Quantum Uncertainty and Interferometric Power by Metric Adjusted Skew Information