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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Length of ray-images under conformal maps


Author: V. Karunakaran
Journal: Proc. Amer. Math. Soc. 87 (1983), 289-294
MSC: Primary 30C45; Secondary 30C35
MathSciNet review: 681836
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Abstract: Let $ w = f(z)$ be regular and univalent in $ \vert z\vert < 1$ with $ f(0) = 0$. Suppose that $ f$ maps the unit disc onto a domain $ D$. Let $ l(r,\theta )$ be the length of the image curve of the ray joining $ z = 0$ to $ z = r{e^{i\theta }}$ in $ D$ and $ A(r) = \operatorname{Sup}[{\left\vert {f(r{e^{i\theta }})} \right\vert^{ - 1}}l(r,\theta )]$ where the supremum is taken over all starlike functions. In this paper we show that $ A(r) \leqslant (1 + r)$.


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DOI: https://doi.org/10.1090/S0002-9939-1983-0681836-9
Keywords: Starlike, ray-image
Article copyright: © Copyright 1983 American Mathematical Society