On Exponentially Concave Functions and Their Impact in Information Theory

Concave functions play a central role in optimization. So-called exponentially concave functions are of similar importance in information theory. In this paper, we comprehensively discuss mathematical properties of the class of exponentially concave functions, like closedness under linear and convex combination and relations to quasi-, Jensen- and Schur-concavity. Information theoretic quantities such as self-information and (scaled) entropy are shown to be exponentially concave. Furthermore, new inequalities for the Kullback-Leibler divergence, for the entropy of mixture distributions, and for mutual information are derived.