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Transactions of the American Mathematical Society
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
MathSciNet review: 684508
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Abstract: Let $ f$ be a conformal map of $ R = \{w = u + iv \in {\mathbf{C}}\vert{\varphi _0}(u) < v < {\varphi _1}(u)\} $ onto $ S = \{z = x + iy \in {\mathbf{C}}\vert < 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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DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0684508-4
PII: S 0002-9947(1983)0684508-4
Article copyright: © Copyright 1983 American Mathematical Society