On a property of harmonic measure on simply connected domains

TL;DR

This paper investigates the relationship between two harmonic measure functions in simply connected domains, providing counterexamples to a conjecture and identifying conditions under which the relationship holds, including the exact constant for starlike domains.

Contribution

The paper demonstrates that the expected inequality between harmonic measures does not always hold, but under certain geometric conditions, it does, and it determines the optimal constant for starlike domains.

Findings

Counterexamples show the inequality does not always hold.

Geometric conditions ensure the inequality holds.

Optimal constant found for starlike 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.