On inequalities for moments and the covariance of monotone functions

Intuition based on the usual interpretation of the covariance of two random variables suggests that the inequality cov[f (X), g (X)] >= 0 should hold for any random variable X and any two increasing functions f and g. The inequality holds indeed, but a proof is hard to find in the literature. In this paper we provide an elementary proof of a more general inequality for moments and we present several applications in actuarial mathematics. (C) 2014 Elsevier B.V. All rights reserved.