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Transactions of the American Mathematical Society

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Univalence criteria in multiply-connected domains


Author: Brad G. Osgood
Journal: Trans. Amer. Math. Soc. 260 (1980), 459-473
MSC: Primary 30C55; Secondary 30C60
DOI: https://doi.org/10.1090/S0002-9947-1980-0574792-7
MathSciNet review: 574792
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Abstract: Theorems due to Nehari and Ahlfors give sufficient conditions for the univalence of an analytic function in relation to the growth of its Schwarzian derivative. Nehari's theorem is for the unit disc and was generalized by Ahlfors to any simply-connected domain bounded by a quasiconformal circle. In both cases the growth is measured in terms of the hyperbolic metric of the domain. In this paper we prove a corresponding theorem for finitely-connected domains bounded by points and quasiconformal circles. Metrics other than the hyperbolic metric are also considered and similar results are obtained.


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  • [1] L. V. Ahlfors, Quasiconformal reflections, Acta Math. 109 (1963), 291-301. MR 0154978 (27:4921)
  • [2] L. V. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975-978. MR 0148896 (26:6393)
  • [3] A. F. Beardon and F. W. Gehring, Schwarzian derivatives, the Poincaré metric and the kernel function, Comment. Math. Helv. (to appear). MR 569245 (81c:30020)
  • [4] A. F. Beardon and Ch. Pommerenke, The Poincaré metric of plane domains, J. London Math. Soc. (2) 18 (1978), 475-483. MR 518232 (80a:30020)
  • [5] P. R. Garabedian and M. Schiffer, Identities in the theory of conformal mappings, Trans. Amer. Math. Soc. 65 (1949), 187-238. MR 0028944 (10:522d)
  • [6] F. W. Gehring, Univalent functions and the Schwarzian derivative, Comment. Math. Helv. 52 (1977), 561-572. MR 0457701 (56:15905)
  • [7] F. W. Gehring and J. Väisälä, The coefficients of quasiconformality in space, Acta Math. 114 (1965), 1-70. MR 0180674 (31:4905)
  • [8] E. Hille, Remarks on a paper by Z. Nehari, Bull. Amer. Math. Soc. 55 (1949), 552-553. MR 0030000 (10:697a)
  • [9] W. Kraus, Über den Zusammenhang einiger Charakteristiken eines einfach zusammenhängenden Bereiches mit der Kreisabbildung, Mitt. Math. Sem. Giessen 21 (1932).
  • [10] O. Lehto, Domain constants associated with the Schwarzian derivative, Comment. Math. Helv. 52 (1977), pp. 603-610. MR 0457703 (56:15907)
  • [11] -, Quasiconformal mappings in the plane, Lecture Notes 14, Univ. of Maryland, 1975.
  • [12] O. Lehto and K. I. Virtanen, Quasiconformal mappings in the plane, Springer-Verlag, Berlin and New York, 1973. MR 0344463 (49:9202)
  • [13] Z. Nehari, The Schwarzian derivative and schlicht functions, Bull. Amer. Math. Soc. 55 (1949), 545-551. MR 0029999 (10:696e)
  • [14] G. Springer, Fredholm eigenvalues and quasiconformal mappings, Acta Math. 111 (1964), 121-142. MR 0161976 (28:5178b)
  • [15] K. Strebel, On the maximal dilatation of quasiconformal mappings, Proc. Amer. Math. Soc. 6 (1953), 903-909. MR 0073702 (17:473d)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1980-0574792-7
Keywords: Schwarzian derivative, hyperbolic metric, quasiconformal circle, kernel function
Article copyright: © Copyright 1980 American Mathematical Society

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