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Transactions of the American Mathematical Society

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



Quasiconformal mappings and Schwarz's lemma

Author: Peter J. Kiernan
Journal: Trans. Amer. Math. Soc. 148 (1970), 185-197
MSC: Primary 30.47
MathSciNet review: 0255807
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Abstract: In this paper, K quasiconformal maps of Riemann surfaces are investigated. A theorem, which is similar to Schwarz's lemma, is proved for a certain class of K quasiconformal maps. This result is then used to give elementary proofs of theorems concerning K quasiconformal maps. These include Schottky's lemma, Liouville's theorem, and the big Picard theorem. Some of Huber's results on analytic self-mappings of Riemann surfaces are also generalized to the K quasiconformal case. Finally, as an application of the Schwarz type theorem, a geometric proof of a special case of Moser's theorem is given.

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Keywords: Quasiconformal, Schwarz's lemma, Poincaré-Bergman metric, Kobayashi pseudo-distance, distance decreasing, Picard theorem, harmonic mappings
Article copyright: © Copyright 1970 American Mathematical Society