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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On entire conformal mappings of simply connected regions


Author: Roy W. Pengra
Journal: Proc. Amer. Math. Soc. 50 (1975), 249-254
MSC: Primary 30A30
MathSciNet review: 0385080
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Abstract: Let $ \Omega $ be a simply connected proper neighborhood of the origin in the complex plane. Let $ \phi $ be a one to one conformal mapping of the unit disk onto $ \Omega $, with $ \phi (0) = 0$. The most general one to one conformal mapping of $ \Omega $ onto itself which fixes the origin has the form $ {f_\lambda }(z) = \phi (\lambda {\phi ^{ - 1}}(z))$ where $ \vert\lambda \vert = 1$. It is shown that the set of $ \lambda $ for which $ {f_\lambda }$ is entire and nonlinear has Lebesgue measure zero on the unit circle. The proof depends in part upon properties of solutions, $ \phi $, of the functional equation $ f(\phi (w)) = \phi (\lambda w)$, where $ f$ is an entire, nonlinear function, $ f(0) = 0$ and $ \vert\lambda \vert = 1$.


References [Enhancements On Off] (What's this?)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0385080-X
PII: S 0002-9939(1975)0385080-X
Article copyright: © Copyright 1975 American Mathematical Society