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Mapping properties of analytic functions on the unit disk

Author: Alexander Yu. Solynin
Journal: Proc. Amer. Math. Soc. 136 (2008), 577-585
MSC (2000): Primary 30C55, 30F45
Published electronically: November 3, 2007
MathSciNet review: 2358498
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Abstract: Let $ f$ be analytic on the unit disk $ \mathbb{D}$ with $ f(0)=0$. In 1989, D. Marshall conjectured the existence of the universal constant $ r_0>0$ such that $ f(r_0\mathbb{D})\subset \mathbb{D}_M:=\{w:\,\vert w\vert<M\}$ whenever the area, counting multiplicity, of a portion of $ f(\mathbb{D})$ over $ \mathbb{D}_M$ is $ <\pi M^2$. Recently, P. Poggi-Corradini (2007) proved this conjecture with an unspecified constant by the method of extremal metrics. In this note we show that such a universal constant $ r_0$ exists for a much larger class consisting of analytic functions omitting two values of a certain doubly-sheeted Riemann surface. We also find a numerical value, $ r_0=.03949\ldots$, which is sharp for the problem in this larger class but is not sharp for Marshall's problem.

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Additional Information

Alexander Yu. Solynin
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Box 41042, Lubbock, Texas 79409

Keywords: Analytic function, growth theorem, hyperbolic metric
Received by editor(s): October 26, 2006
Published electronically: November 3, 2007
Additional Notes: This research was supported in part by NSF grant DMS-0525339
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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