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Concentration of area in half-planes
Author(s):
Roger
W.
Barnard;
Clint
Richardson;
Alexander
Yu.
Solynin
Journal:
Proc. Amer. Math. Soc.
133
(2005),
2091-2099.
MSC (2000):
Primary 30C70, 30E20
Posted:
January 31, 2005
MathSciNet review:
2137876
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Abstract:
For the standard class  of normalized univalent functions  analytic in the unit disk  , we consider a problem on the minimal area of the image  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.
References:
-
- 1.
- R. W. Barnard, K. Pearce, and A. Yu. Solynin, An isoperimetric inequality for logarithmic capacity. Annales AcademiæScientiarum Fennicæ. Mathematica 27 (2002), 419-436. MR 1922198 (2003g:30039)
- 2.
- R. W. Barnard and A. Yu. Solynin, Local variations and minimal area problem for Carathéodory functions. Indiana U. Math. J., 53 (2004), no. 1, 135-167.
- 3.
- V. N. Dubinin, Symmetrization in geometric theory of functions of a complex variable. Uspehi Mat. Nauk 49 (1994), 3-76 (in Russian); English translation in Russian Math. Surveys 49: 1 (1994), 1-79. MR 1307130 (96b:30054)
- 4.
- W. K. Hayman, Multivalent Functions. Second edition. Cambridge Tracts in Mathematics, 110. Cambridge Univ. Press, Cambridge, 1994. MR 1310776 (96f:30003)
- 5.
- Ch. Pommerenke, Boundary Behaviour of Conformal Maps. Springer-Verlag, 1992. MR 1217706 (95b:30008)
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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:
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
Posted:
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
Copyright of article:
Copyright
2005,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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