Saturable Generalizations of Jensen's Inequality

André M. Timpanaro

Published 2021-02-24Version 1

Jensen's inequality can be thought as answering the question of how knowledge of $\mathbb{E}(X)$ allows us to bound $\mathbb{E}(f(X))$ in the cases where $f$ is either convex or concave. Many generalizations have been proposed, which boil down to how additional knowledge allows us to sharpen the inequality. In this work, we investigate the question of how knowledge about expectations $\mathbb{E}(f_i(X))$ of a random vector $X$ translate into bounds for $\mathbb{E}(g(X))$. Our results show that there is a connection between this problem and properties of convex hulls, allowing us to rewrite it as an optimization problem. The results of these optimization problems not only arrive at sharp bounds for $\mathbb{E}(g(X))$ but in some cases also yield discrete probability measures where equality holds. We develop both analytical and numerical approaches for finding these bounds and also study in more depth the case where the known information are the average and variance, for which some analytical results can be obtained.