The problem of finding necessary and sufficient conditions for the existence of a Kähler-Einstein metric on a Fano manifold has attracted much attention. Fundamental work done in the ’80s related this question to certain energy functionals defined on the infinite-dimensional space of Kähler metrics. In the mid ’90s Tian provided a first characterization of Kähler-Einstein manifolds in terms of two such functionals. In this talk we present a generalization of this analytic characterization to a family of energy functionals defined by Chen and Tian. We also show how our arguments provide a new proof of the classical Moser-Trudinger-Onofri inequality on the sphere and how they can be used to formulate an extension of it for higher-dimensional manifolds, extending the work of Ding and Tian.