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St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)



A minimal area problem for nonvanishing functions

Authors: R. W. Barnard, C. Richardson and A. Yu. Solynin
Original publication: Algebra i Analiz, tom 18 (2006), nomer 1.
Journal: St. Petersburg Math. J. 18 (2007), 21-36
MSC (2000): Primary 30C70, 30E20
Published electronically: November 27, 2006
MathSciNet review: 2225212
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Abstract | References | Similar Articles | Additional Information

Abstract: The minimal area covered by the image of the unit disk is found for nonvanishing univalent functions normalized by the conditions $ f(0) = 1$, $ f'(0) = \alpha$. Two different approaches are discussed, each of which contributes to the complete solution of the problem. The first approach reduces the problem, via symmetrization, to the class of typically real functions, where the well-known integral representation can be employed to obtain the solution upon a priori knowledge of the extremal function. The second approach, requiring smoothness assumptions, leads, via some variational formulas, to a boundary value problem for analytic functions, which admits an explicit solution.

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Additional Information

R. W. Barnard
Affiliation: Department of Mathematics and Statistics, Texas Tech. University, Box 41042, Lubbock, Texas 79409

C. Richardson
Affiliation: Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, Texas 75962

A. Yu. Solynin
Affiliation: Department of Mathematics and Statistics, Texas Tech. University, Box 41042, Lubbock, Texas 79409

Keywords: minimal area problem, nonvanishing analytic function, typically real function, symmetrization
Received by editor(s): August 15, 2005
Published electronically: November 27, 2006
Additional Notes: The third author’s research was partially supported by NSF (grant DMS–0412908).
Article copyright: © Copyright 2006 American Mathematical Society

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