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The length of ray-images under starlike mappings

Published online by Cambridge University Press:  26 February 2010

R. R. Hall
R. R. Hall
Affiliation:
The Department of Mathematics, The University of York, England
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Extract

Let w = f(z) be regular and schlicht for |z|, and f (0) = 0.

Suppose that f maps the unit disc {z : |z| < 1} onto a domain D starlike with respect to w = 0. Let C(r, θ) be the image in D of the ray joining z = 0 to z = reiθ, and let

be its length. Sheil–Small [1] proved that l(r, θ) < (1 + log 4) | f (reiθ)|, and conjectured the following result, which it is my aim to prove in this paper.

Type
Research Article
Information
Mathematika , Volume 23 , Issue 2 , December 1976 , pp. 147 - 150
Copyright
Copyright © University College London 1976

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References

1.Sheil-Small, T.. “Some conformal mapping inequalities for starlike and convex functions”, J. London Math. Soc. (2), 1 (1969), 577–587.Google Scholar
2.Zygmund, A.. Trigonometric Series (Cambridge, 1968).Google Scholar