Hyperbolic distance and membership of conformal maps in the Hardy space

Abstract

Let $\psi$ be a conformal map of the unit disk $\mathbb{D}$ 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 ${H^p}\left( \mathbb{D} \right)$ denotes the classical Hardy space and $d_\mathbb{D} {\left( {0,{F_\alpha }} \right)}$ denotes the hyperbolic distance between $0$ and $F_\alpha$ in $\mathbb{D}$, we prove that $\psi$ belongs to ${H^p}\left( \mathbb{D} \right)$ if and only if \[\int_0^{ + \infty } {{\alpha ^{p - 1}}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_\alpha }} \right)}}d\alpha } < + \infty .\] This result answers a question posed by P. Poggi-Corradini.