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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Concentration of area in half-planes


Authors: Roger W. Barnard, Clint Richardson and Alexander Yu. Solynin
Journal: Proc. Amer. Math. Soc. 133 (2005), 2091-2099
MSC (2000): Primary 30C70, 30E20
Published electronically: January 31, 2005
MathSciNet review: 2137876
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Abstract | References | Similar Articles | Additional Information

Abstract: For the standard class $S$ of normalized univalent functions $f$analytic in the unit disk $\mathbb{U}$ , we consider a problem on the minimal area of the image $f(\mathbb{U})$ concentrated in any given half-plane. This question is related to a well-known problem posed by A. W. Goodman in 1949 that regards minimizing area covered by analytic univalent functions under certain geometric constraints. An interesting aspect of this problem is the unexpected behavior of the candidates for extremal functions constructed via geometric considerations.


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Additional Information

Roger W. Barnard
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409
Email: barnard@math.ttu.edu

Clint Richardson
Affiliation: Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, Texas 75962
Email: crichardson@sfasu.edu

Alexander Yu. Solynin
Affiliation: Department of Mathematics and Statistics, Texas Tech University, Lubbock, Texas 79409
Email: solynin@math.ttu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-07775-0
PII: S 0002-9939(05)07775-0
Keywords: Minimal area problem, univalent function, local variation, symmetrization
Received by editor(s): April 5, 2002
Received by editor(s) in revised form: March 22, 2004
Published electronically: January 31, 2005
Additional Notes: The research of the second author was supported in part by the Summer Dissertation/Thesis Award of the Graduate School of Texas Tech University
The research of the third author was supported in part by the Russian Foundation for Basic Research, grant no. 00-01-00118a.
Communicated by: Juha M. Heinonen
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.