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Algebra i Analiz, 2006, Volume 18, Issue 1, Pages 34–54 (Mi aa59)  

This article is cited in 3 scientific papers (total in 3 papers)

Research Papers

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

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.

Keywords: minimal area problem, nonvanishing analytic function, typically real function, symmetrization

Full text: PDF file (204 kB)
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English version:
St. Petersburg Mathematical Journal, 2007, 18:1, 21–36

Bibliographic databases:

MSC: 30C70, 30E20
Received: 15.08.2005
Language:

Citation: R. W. Barnard, C. Richardson, A. Yu. Solynin, “A minimal area problem for nonvanishing functions”, Algebra i Analiz, 18:1 (2006), 34–54; St. Petersburg Math. J., 18:1 (2007), 21–36

Citation in format AMSBIB
\Bibitem{BarRicSol06}
\by R.~W.~Barnard, C.~Richardson, A.~Yu.~Solynin
\paper A minimal area problem for nonvanishing functions
\jour Algebra i Analiz
\yr 2006
\vol 18
\issue 1
\pages 34--54
\mathnet{http://mi.mathnet.ru/aa59}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2225212}
\zmath{https://zbmath.org/?q=an:1122.30017}
\elib{http://elibrary.ru/item.asp?id=9212598}
\transl
\jour St. Petersburg Math. J.
\yr 2007
\vol 18
\issue 1
\pages 21--36
\crossref{https://doi.org/10.1090/S1061-0022-06-00941-1}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Beneteau C., Khavinson D., “A survey of certain extremal problems for non-vanishing analytic functions”, Complex and Harmonic Analysis, 2007, 45–61  mathscinet  zmath  isi
    2. Barnard R.W., Pearce K., Solynin A.Yu., “Iceberg-Type Problems: Estimating Hidden Parts of a Continuum From the Visible Parts”, Math. Nachr., 285:17-18 (2012), 2042–2058  crossref  mathscinet  zmath  isi  elib  scopus
    3. Barnard R.W., Lochman M.H., Solynin A.Yu., “Convex Icebergs and Sectorial Starlike Functions”, Comput. Methods Funct. Theory, 13:4 (2013), 635–682  crossref  mathscinet  zmath  isi  elib  scopus
  • Алгебра и анализ St. Petersburg Mathematical Journal
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