Some Moment Inequalities Involving Generalized Convex Functions

A standard result known as Chebychev's inequality asserts that any two non-decreasing functions of a real random variable have non-negative correlation. We show that any two convex functions have non-negative partial correlation, after adjusting for the linear term. This is the simplest in a large family of new results, some of which involve concepts of higher-order convexity. However we have not succeeded in obtaining multivariate versions of these results.