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The problem of finding necessary and sufficient conditions for the existence
of a K\"{a}hler-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 K\"{a}hler metrics. In the mid '90s
Tian provided a first characterization of K\"{a}hler-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.
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