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 | References | Similar Articles | Additional Information

Abstract: Let be analytic on the unit disk with . In 1989, D. Marshall conjectured the existence of the universal constant such that whenever the area, counting multiplicity, of a portion of over is . 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 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, , 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

Email:
alex.solynin@ttu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-07-09080-6

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.