  We prove Aubin's "Hypothese fondamentale" concerning the existence of
Moser-Trudinger type inequalities on any integral compact K\"ahler manifold X.
In the case of the anti-canonical class on a Fano manifold the constants in the
inequalities are shown to only depend on the dimension of X (but there are
counterexamples to the precise value proposed by Aubin). In the different
setting of pseudoconvex domains in complex space we also obtain a quasi-sharp
version of the inequalities and relate it to Brezis-Merle type inequalities.
The inequalities are shown to be sharp for S^{1}-invariant functions on the
unit-ball. We give applications to existence and blow-up of solutions to
complex Monge-Amp\`ere equations of mean field (Liouville) type.
