Zero subsequences for classes of holomorphic functions: stability and the entropy of arcwise connectedness. I

For a domain $\Omega$ in a 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 about “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.