On a property of harmonic measure on simply connected domains

Abstract

Let $D \subset \mathbb{C}$ be a domain with $0 \in D$. For $R>0$, let ${{\hat \omega }_D}\left( {R} \right)$ denote the harmonic measure of $ D \cap \left\{ {\left| z \right| = R} \right\}$ at $0$ with respect to the domain $ D \cap \left\{ {\left| z \right| < R} \right\} $ and ${\omega_D}\left( {R} \right)$ denote the harmonic measure of $\partial D \cap \left\{ {\left| z \right| \ge R} \right\}$ at $0$ with respect to $D$. The behavior of the functions ${\omega_D}$ and ${{\hat \omega }_D}$ near $\infty$ determines (in some sense) how large $D$ is. However, it is not known whether the functions ${\omega_D}$ and ${{\hat \omega }_D}$ always have the same behavior when $R$ tends to $\infty$. Obviously, ${\omega_D}\left( {R} \right) \le {{\hat \omega }_D}\left( {R} \right)$ for every $R>0$. Thus, the arising question, first posed by Betsakos, is the following: Does there exist a positive constant $C$ such that for all simply connected domains $D$ with $0 \in D$ and all $R>0$, \[{\omega_D}\left( {R} \right) \ge C{{\hat \omega }_D}\left( {R} \right)? \] In general, we prove that the answer is negative by means of two different counter-examples. However, under additional assumptions involving the geometry of $D$, we prove that the answer is positive. We also find the value of the optimal constant for starlike domains.