  We prove that admissible functions for Fubini-Study metrics on the complex
projective space $P_{m}C$, of complex dimension $m$, invariant by a convenient
automorphisms group, are lower bounded by a function going to minus infinity on
the boundary of usual charts of $P_{m}C$. A similar lower bound holds on some
projective manifolds. This gives an optimal constant in a Hormander type
inequality on these manifolds, which allows us to establish the existence of
Einstein-Kahler metrics on them.
