Counting eigenvalues of Schr\"odinger operator with complex fast decreasing potential

We give a sharp estimate of the number of zeros of analytic functions in the unit disc belonging to analytic quasianalytic Carleman--Gevrey classes. As an application, we estimate the number of the eigenvalues for discrete Schr\"odinger operators with rapidly decreasing complex-valued potentials, and, more generally, for non-symmetric Jacobi matrices.