  We prove sharp inequalities for determinants of Toeplitz operators and
twisted Laplace operators on the two-sphere, generalizing the
Moser-Trudinger-Onofri inequality. In particular a sharp version of conjectures
of Gillet-Soule and Fang motivated by Arakelov geometry is obtained;
applications to SU(2)-invariant determinantal random point processes on the
two-sphere are also discussed. The inequalities are obtained as corollaries of
a general theorem about the maximizers of a certain non-local functional
defined on the space of all positively curved Hermitian metrics on an ample
line bundle over a compact complex manifold. This functional is an adjoint
version, introduced by Berndtsson, of Donaldson's L-functional and generalizes
the Ding-Tian functional whose critical points are Kahler-Einstein metrics. In
particular, new proofs of some results in Kahler geometry are also obtained,
including a lower bound on Mabuchi's K-energy and the uniqueness result for
Kahler-Einstein metrics on Fano manifolds of Bando-Mabuchi.
