  Yau conjectured that a Fano manifold admits a Kahler-Einstein metric if and
only if it is stable in the sense of geometric invariant theory. There has been
much progress on this conjecture by Tian, Donaldson and others. The Mabuchi
energy functional plays a central role in these ideas.
  We study the E_k functionals introduced by X.X. Chen and G. Tian which
generalize the Mabuchi energy. We show that if a Fano manifold admits a
Kahler-Einstein metric then the functional E_1 is bounded from below, and,
modulo holomorphic vector fields, is proper. This answers affirmatively a
question raised by Chen. We show in fact that E_1 is proper if and only if
there exists a Kahler-Einstein metric, giving a new analytic criterion for the
existence of this canonical metric, with possible implications for the study of
stability.
  We also show that on a Fano Kahler-Einstein manifold all of the functionals
E_k are bounded below on the space of metrics with nonnegative Ricci curvature.
