On the Distribution of Zero Sets of Holomorphic Functions. IV. A Criterion

Abstract

Let $D$ be a proper domain in the extended complex plane ${\mathbb{C}}_{\infty }:=\mathbb{C}\cup \{\infty \}$, $M=M_{+}-M_{-}\not\equiv \pm \infty$ be a difference of non-trivial subharmonic functions $M_{\pm }\not\equiv \mp \infty$ on $D$, $\text{Hol}(D,M)$ be the class of holomorphic function $f$ on $D$ satisfying $|f|\le {\text{const}}_f\exp M$ om $D$, $\mathsf{Z}\subset D$ be a sequence of points in $D$ without limits points in $D$. We give a complete description of the conditions under which the sequence $\mathsf{Z}$ is a sequence of all zeros for some nonzero function $f\in \text{Hol}(D,M)$.

7 pages, in Russian