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ЖУРНАЛЫ // Алгебра и анализ // Архив

Алгебра и анализ, 2006, том 18, выпуск 1, страницы 34–54 (Mi aa59)

Эта публикация цитируется в 3 статьях

Статьи

A minimal area problem for nonvanishing functions

R. W. Barnarda, C. Richardsonb, A. Yu. Solynina

a Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX
b Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, TX

Аннотация: 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.

Ключевые слова: minimal area problem, nonvanishing analytic function, typically real function, symmetrization.

MSC: 30C70, 30E20

Поступила в редакцию: 15.08.2005

Язык публикации: английский


 Англоязычная версия: St. Petersburg Mathematical Journal, 2007, 18:1, 21–36

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