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An extension of the Ahlfors distortion theorem

Published online by Cambridge University Press:  24 October 2008

F. Huckemann
F. Huckemann
Affiliation:
Queens' College†Cambridge
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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

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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
(2)Collingwood, E. F. and Cartwright, M. L.Boundary theorems for a function mero-morphic in the unit circle. Acta Math., Stockh., 87 (1952), 83–146.CrossRefGoogle Scholar
(3)Teichmüller, O.Untersuchungen über konforme und quasikonforme Abbildung. Dtsch. Math. 3 (1938), 621–78.Google Scholar