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Proceedings of the American Mathematical Society

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

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-07-09080-6
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 $f(r_0\mathbb {D})\subset \mathbb {D}_M:=\{w: |w|<M\}$ 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
MR Author ID: 206458
Email: alex.solynin@ttu.edu

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.