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.

24 pages, 23 figures