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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Mapping properties of analytic functions on the unit disk
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by Alexander Yu. Solynin PDF
Proc. Amer. Math. Soc. 136 (2008), 577-585 Request permission

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
  • 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
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2358498