  We consider the number of zeros of holomorphic functions in a bounded domain
that depend on a small parameter and satisfy an exponential upper bound near
the boundary of the domain and similar lower bounds at finitely many points
along the boundary. Roughly the number of such zeros is $(2\pi h)^{-1}$ times
the integral over the domain of the laplacian of the exponent of the dominating
exponential. Such results have already been obtained by M. Hager and by Hager
and the author and they are of importance when studying the asymptotic
distribution of eigenvalues of elliptic operators with small random
perturbations. In this paper we generalize these results and arrive at
geometrically natural statements and natural remainder estimates.
