Proceedings of the American Mathematical Society

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



Mapping properties of analytic functions on the disk

Author: Pietro Poggi-Corradini
Journal: Proc. Amer. Math. Soc. 135 (2007), 2893-2898
MSC (2000): Primary 30C55
Published electronically: May 8, 2007
MathSciNet review: 2317966
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Abstract: There is a universal constant $ 0<r_0<1$ with the following property. Suppose that $ f$ is an analytic function on the unit disk $ \mathbb{D}$, and suppose that there exists a constant $ M>0$ so that the Euclidean area, counting multiplicity, of the portion of $ f(\mathbb{D})$ which lies over the disk $ D(f(0),M)$, centered at $ f(0)$ and of radius $ M$, is strictly less than the area of $ D(f(0),M)$. Then $ f$ must send $ r_0\overline{\mathbb{D}}$ into $ D(f(0),M)$. This answers a conjecture of Don Marshall.

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Pietro Poggi-Corradini
Affiliation: Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506

Received by editor(s): January 3, 2006
Received by editor(s) in revised form: June 1, 2006
Published electronically: May 8, 2007
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2007 American Mathematical Society