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An extension of the Ahlfors distortion theorem
Published online by Cambridge University Press: 24 October 2008
Extract
1. The conformal mapping of a strip domain in the z-plane on to a parallel strip— parallel, say, to the real axis of the ζ ( = ξ + iμ)-plane—brings about a certain distortion. More precisely: consider a cross-cut on the line ℜz = c joining the two sides of the frontier of the strip domain (in these introductory remarks we suppose for simplicity that there is only one such cross-cut on that line), and denote by ξ1(c) and ξ2(c) the lower and upper bounds of ξ on the image in the ζ-plane. The theorem of Ahlfors (1), now classical, states that
provided that
where a is the width of the parallel strip and θ(c) the length of the cross-cut.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 50 , Issue 2 , April 1954 , pp. 261 - 265
- Copyright
- Copyright © Cambridge Philosophical Society 1954
References
REFERENCES
(1)Ahlfors, L.Untersuchungen zur Theorie der konformen Abbildung und der ganzen Funktionen. Acta Soc. Sci.fenn., N.S., 1, 9 (1930), 1–40.Google Scholar
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(3)Teichmüller, O.Untersuchungen über konforme und quasikonforme Abbildung. Dtsch. Math. 3 (1938), 621–78.Google Scholar