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.