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Transactions of the American Mathematical Society

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

 

 

Univalence criteria in multiply-connected domains


Author: Brad G. Osgood
Journal: Trans. Amer. Math. Soc. 260 (1980), 459-473
MSC: Primary 30C55; Secondary 30C60
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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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