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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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


Authors: Burton Rodin and S. E. Warschawski
Journal: Trans. Amer. Math. Soc. 276 (1983), 281-288
MSC: Primary 30C20; Secondary 30C35
DOI: https://doi.org/10.1090/S0002-9947-1983-0684508-4
MathSciNet review: 684508
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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.


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Article copyright: © Copyright 1983 American Mathematical Society