A necessary and sufficient condition for the asymptotic version of Ahlfors’ distortion property

Abstract:

Let $f$ be a conformal map of $R = \{w = u + iv \in {\mathbf {C}}|{\varphi _0}(u) < v < {\varphi _1}(u)\}$ onto $S = \{z = x + iy \in {\mathbf {C}}|0 < y < 1\}$ where the ${\varphi _j} \in {C^0}( - \infty ,\infty )$ and $\operatorname {Re} f(w) \to \pm \infty$ as $\operatorname {Re} w \to \pm \infty$. There are well-known results giving conditions on $R$ sufficient for the distortion property $\operatorname {Re} f(u + iv) = \int _0^u ({\varphi _1} - {\varphi _0})^{- 1}du + {\text {const}}. + o(1)$, where $o(1) \to 0$ as $u \to + \infty$. In this paper the authors give a condition on $R$ which is both necessary and sufficient for $f$ to have this property.