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 \[\int_0^{ + \infty } {α^{p - 1}{e^{ - {d_{\mathbb{D}}}\left( {0,{F_α}} \right)}}dα} < + \infty .\] This result answers a question posed by P. Poggi-Corradini.

4 pages