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.
Categories: math.CV