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 | References | Similar Articles | Additional Information

Abstract: There is a universal constant with the following property. Suppose that is an analytic function on the unit disk , and suppose that there exists a constant so that the Euclidean area, counting multiplicity, of the portion of which lies over the disk , centered at and of radius , is strictly less than the area of . Then must send into . This answers a conjecture of Don Marshall.

**[Ahl73]**Lars V. Ahlfors,*Conformal invariants: topics in geometric function theory*, McGraw-Hill Book Co., New York-Düsseldorf-Johannesburg, 1973. McGraw-Hill Series in Higher Mathematics. MR**0357743****[Mar89]**Donald E. Marshall,*A new proof of a sharp inequality concerning the Dirichlet integral*, Ark. Mat.**27**(1989), no. 1, 131–137. MR**1004727**, 10.1007/BF02386365**[MSZ03]**Jan Malý, David Swanson, and William P. Ziemer,*The co-area formula for Sobolev mappings*, Trans. Amer. Math. Soc.**355**(2003), no. 2, 477–492 (electronic). MR**1932709**, 10.1090/S0002-9947-02-03091-X**[Oht70]**Makoto Ohtsuka.*Dirichlet problem, extremal length, and prime ends*.

Van Nostrand, 1970.**[Ric93]**Seppo Rickman,*Quasiregular mappings*, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR**1238941**

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Additional Information

**Pietro Poggi-Corradini**

Affiliation:
Department of Mathematics, Cardwell Hall, Kansas State University, Manhattan, Kansas 66506

Email:
pietro@math.ksu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-07-08823-5

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