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Transactions of the American Mathematical Society

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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}}\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.


References [Enhancements On Off] (What's this?)

  • [1] L. V. Ahlfors, Untersuchungen zur Theorie der konformen Abbildungen und der ganzen Funktionen, Ann. Acad. Sci. Fenn. Ser. AI Math. 9 (1930), 1-40.
  • [2] B. G. Eke, Remarks on Ahlfors' distortion theorem, J. Analyse Math. 19 (1967), 97-134. MR 0215971 (35:6806)
  • [3] J. Lelong-Ferrand, Représentation conforme et transformations à intégrale de Dirichlet borné, Gauthier-Villars, Paris, 1955. MR 0069895 (16:1096b)
  • [4] J. A. Jenkins and K. Oikawa, On results of Ahlfors and Hayman, Illinois J. Math. 15 (1971), 664-671. MR 0296271 (45:5332)
  • [5] -, On Ahlfors' "second fundamental inequality", Proc. Amer. Math. Soc. 62 (1977), 266-270. MR 0437732 (55:10655)
  • [6] B. Rodin, The method of extremal length, Bull. Amer. Math. Soc. 80 (1974), 587-606. MR 0361048 (50:13494)
  • [7] B. Rodin and S. E. Warschawski, On conformal mapping of $ L$-strips, J. London Math. Soc. (2) 11 (1975), 301-307. MR 0399437 (53:3281)
  • [8] -, Extremal length and the boundary behavior of conformal mappings, Ann. Acad. Sci. Fenn. Ser. AI Math. 2 (1976), 467-500. MR 0466516 (57:6394)
  • [9] -, Extremal length and univalent functions III. Consequences of the Ahlfors distortion property, Bull. Inst. Math. Acad. Sinica 6 (1978), 583-597. MR 528670 (81k:30013)
  • [10] -, Extremal length and univalent functions II. Integral estimates of strip mappings, J. Math. Soc. Japan 31 (1979), 87-99. MR 519038 (81k:30012)
  • [11] S. E. Warschawski, On conformal mapping of infinite strips, Trans. Amer. Math. Soc. 51 (1942), 280-335. MR 0006583 (4:9b)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0684508-4
Article copyright: © Copyright 1983 American Mathematical Society

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