A minimal area problem for nonvanishing functions
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- by R. W. Barnard, C. Richardson and A. Yu. Solynin
- St. Petersburg Math. J. 18 (2007), 21-36
- DOI: https://doi.org/10.1090/S1061-0022-06-00941-1
- Published electronically: November 27, 2006
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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.References
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Bibliographic Information
- R. W. Barnard
- Affiliation: Department of Mathematics and Statistics, Texas Tech. University, Box 41042, Lubbock, Texas 79409
- MR Author ID: 31355
- Email: roger.w.barnard@ttu.edu
- C. Richardson
- Affiliation: Department of Mathematics and Statistics, Stephen F. Austin State University, Nacogdoches, Texas 75962
- Email: crichardson@sfasu.edu
- A. Yu. Solynin
- Affiliation: Department of Mathematics and Statistics, Texas Tech. University, Box 41042, Lubbock, Texas 79409
- MR Author ID: 206458
- Email: alex.solynin@ttu.edu
- 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).
- © Copyright 2006 American Mathematical Society
- Journal: St. Petersburg Math. J. 18 (2007), 21-36
- MSC (2000): Primary 30C70, 30E20
- DOI: https://doi.org/10.1090/S1061-0022-06-00941-1
- MathSciNet review: 2225212