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

Abstract:

Let $\Omega$ be a domain in the complex plane $\mathbb C$, $H(\Omega )$ the space of functions holomorphic in $\Omega$, and $\mathscr {P}$ a family of functions subharmonic in $\Omega$. Denote by $H_{\mathscr {P}}(\Omega )$ the class of functions $f\in H(\Omega )$ satisfying $|f(z)|\leq C_f\exp p_f(z)$ for all $z\in \Omega$, where $p_f \in \mathscr {P}$ and $C_f$ is a constant. Conditions are found ensuring that a sequence $\Lambda =\{\lambda _k\} \subset \Omega$ be a subsequence of zeros for various classes $H_{\mathscr {P}}(\Omega )$. As a rule, the results and the method are new already when $\Omega =\mathbb {D}$ is the unit circle and $\mathscr {P}$ is a system of radial majorants $p(z)=p(|z|)$.