  Our aim is to give a version of the Moser-Trudinger inequality in the setting
of complex geometry. As a very particular case, our result already gives a new
Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a
domain in $R^2$. We also deduce a new necessary condition for the complex
Monge-Ampere equation for a given measure on a compact Kahler manifold to admit
a Holder continuous solution.
