Quasiconformal mappings and Schwarz’s lemma

Kiernan, Peter J.

doi:10.1090/S0002-9947-1970-0255807-6

Quasiconformal mappings and Schwarz’s lemma
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by Peter J. Kiernan PDF

Trans. Amer. Math. Soc. 148 (1970), 185-197 Request permission

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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