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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
DOI: https://doi.org/10.1090/S1061-0022-06-00941-1
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.


References [Enhancements On Off] (What's this?)

  • 1. D. Aharonov, C. Bénéteau, D. Khavinson, and H. Shapiro, Extremal problems for nonvanishing functions in Bergman spaces, Selected Topics in Complex Analysis. S. Ya. Khavinson Memorial Volume (V. Eiderman and M. Samokhin, eds.), Oper. Theory Adv. Appl., vol. 158, Birkhäuser, Basel, 2005, pp. 59-86. MR 2147588 (2006i:30047)
  • 2. D. Aharonov, H. S. Shapiro, and A. Yu. Solynin, A minimal area problem in conformal mapping, J. Anal. Math. 78 (1999), 157-176. MR 1714449 (2000j:30040)
  • 3. -, Minimal area problems for functions with integral representation, J. Anal. Math. 98 (2006) (to appear).
  • 4. R. W. Barnard, C. Richardson, and A. Yu. Solynin, Concentration of area in half-planes, Proc. Amer. Math. Soc. 133 (2005), no. 7, 2091-2099. MR 2137876 (2006b:30048)
  • 5. R. W. Barnard and A. Yu. Solynin, Local variations and minimal area problem for Carathéodory functions, Indiana Univ. Math. J. 53 (2004), no. 1, 135-167. MR 2048187 (2005a:30028)
  • 6. V. N. Dubinin, Symmetrization in the geometric theory of functions of a complex variable, Uspekhi Mat. Nauk 49 (1994), no. 1, 3-76; English transl., Russian Math. Surveys 49 (1994), no. 1, 1-79. MR 1307130 (96b:30054)
  • 7. P. Duren, Univalent functions, Grundlehren Math. Wiss., vol. 259, Springer-Verlag, New York, 1983. MR 0708494 (85j:30034)
  • 8. W. K. Hayman, Multivalent functions, 2nd ed., Cambridge Tracts in Math., vol. 110, Cambridge Univ. Press, Cambridge, 1994. MR 1310776 (96f:30003)
  • 9. W. K. Hayman and P. B. Kennedy, Subharmonic functions. Vol. I, London Math. Soc. Monogr., No. 9, Academic Press, London-New York, 1976. MR 0460672 (57:665)
  • 10. Ch. Pommerenke, Boundary behaviour of conformal maps, Grundlehren Math. Wiss., vol. 299, Springer-Verlag, Berlin, 1992. MR 1217706 (95b:30008)

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Additional Information

R. W. Barnard
Affiliation: Department of Mathematics and Statistics, Texas Tech. University, Box 41042, Lubbock, Texas 79409
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
Email: alex.solynin@ttu.edu

DOI: https://doi.org/10.1090/S1061-0022-06-00941-1
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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