Subsequences of zeros for classes of holomorphic functions, their stability, and the entropy of arcwise connectedness. I

Abstract:

For a domain $\Omega$ in the complex plane $\mathbb C$, let $H(\Omega )$ denote the space of functions holomorphic in $\Omega$, and let $\mathscr {P}$ be a family of functions subharmonic in $\Omega$. Denote by $H_{\mathscr {P}}(\Omega )$ the class of $f\in H(\Omega )$ satisfying $|f(z)|\leq C_f\exp p_f(z)$, $z\in \Omega$, where $p_f \in \mathscr {P}$ and $C_f$ is a constant. The paper is aimed at conditions for a set $\Lambda \subset \Omega$ to be included in the zero set of some nonzero function in $H_{\mathscr {P}}(\Omega )$. In the first part, certain preparatory theorems are established concerning “quenching” the growth of a subharmonic function by adding to it a function of the form $\log |h|$, where $h$ is a nonzero function in $H(\Omega )$. The method is based on the balayage of measures and subharmonic functions.