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 $R>0$, \[{ω_D}\left( {R} \right) \ge C{{\hat ω}_D}\left( {R} \right)? \] 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.

arXiv comment22 pages, 13 figures