  We prove that the existence of a Kahler-Einstein metric on a Fano manifold is
equivalent to the properness of the energy functionals defined by Bando, Chen,
Ding, Mabuchi and Tian on the set of Kahler metrics with positive Ricci
curvature. We also prove that these energy functionals are bounded from below
on this set if and only if one of them is. This answers two questions raised by
X.-X. Chen. As an application, we obtain a new proof of the classical
Moser-Trudinger-Onofri inequality on the two-sphere, as well as describe a
canonical enlargement of the space of Kahler potentials on which this
inequality holds on higher-dimensional Fano Kahler-Einstein manifolds.
