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

Abstract

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