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.