  Consider an h-pseudodifferential operator P, whose symbol extends
holomorphically to a tubular neighborhood of the real phase space and converges
sufficiently fast to 1, so that the determinant of P is well-defined. We show
that the modulus of this determinant is asymptotically bounded by an
exponential of the integral of the logarithm of the modulus of the symbol along
a certain complex deformation of the real phase space. Since there are many
possible such deformations, we get a variational problem. The paper is devoted
to the corresponding variational calculus.
