On a relation between harmonic measure and hyperbolic distance on planar domains

Abstract

Let $\psi $ be a conformal map of $\mathbb{D}$ onto an unbounded domain and, for $\alpha >0$, let ${F_\alpha }=\left\{ {z \in \mathbb{D}:\left| {\psi \left( z \right)} \right| = \alpha } \right\}$. If $\omega _\mathbb{D}\left( {0,{F_\alpha }} \right)$ denotes the harmonic measure at $0$ of $F_\alpha $ and $d_\mathbb{D} {\left( {0,{F_\alpha }} \right)}$ denotes the hyperbolic distance between $0$ and $F_\alpha$ in $\mathbb{D}$, then an application of the Beurling-Nevanlinna projection theorem implies that ${\omega _\mathbb{D}}\left( {0,{F_\alpha }} \right) \ge \frac{2}{\pi }{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}$. Thus a natural question, first stated by P. Poggi-Corradini, is the following: Does there exist a positive constant $K$ such that for every $\alpha >0$, ${\omega _\mathbb{D}}\left( {0,{F_\alpha }} \right) \le K{e^{ - {d_\mathbb{D}}\left( {0,{F_\alpha }} \right)}}$? In general, we prove that the answer is negative by means of two different examples. However, under additional assumptions involving the number of components of $F_\alpha$ and the hyperbolic geometry of the domain $\psi \left( \mathbb{D} \right)$, we prove that the answer is positive.