Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A finite capacity analogue of the Koebe one-quarter theorem

Author: Robin Cunningham
Journal: Proc. Amer. Math. Soc. 119 (1993), 869-875
MSC: Primary 30C25
MathSciNet review: 1161400
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A variational method is used to determine the largest disk about the origin covered by the image of every normalized univalent function that maps the unit disk onto a region of prescribed logarithmic capacity (transfinite diameter).

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

  • [1] L. Bieberbach, Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln, S. B. Preuss. Akad. Wiss. 35 (1916), 940-955.
  • [2] P. F. Byrd and M. D. Friedman, Handbook of elliptic integrals of engineers and physicists, Springer-Verlag, Berlin, Göttingen, and Heidelberg, 1954; 2nd edition, 1971. MR 0060642 (15:702a)
  • [3] R. Cunningham, Univalent functions of given transfinite diameter: a maximum modulus problem, Ann. Acad. Sci. Fenn. (to appear). MR 1234733 (95f:30033)
  • [4] P. L. Duren, Univalent functions, Springer-Verlag, Heidelberg and New York, 1983. MR 708494 (85j:30034)
  • [5] P. L. Duren and M. Schiffer, Univalent functions which map onto regions of given transfinite diameter, Trans. Amer. Math. Soc. 323 (1991), 413-428. MR 979964 (92d:30013)
  • [6] G. M. Goluzin, Geometric theory of functions of a complex variable, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, 1952; German transl., Deutscher Verlag, Berlin, 1957; 2nd edition, Izdat. "Nauka", Moscow, 1966; English transl., Amer. Math. Soc., Providence, RI, 1969. MR 0247039 (40:308)
  • [7] I. Gradsteyn and I. Ryzhik, Table of integrals, series and products, Academic Press, New York, 1965; enlarged ed., 1980.
  • [8] E. Hille, Analytic function theory, Vol. II, Ginn and Co., Boston, MA, 1962; second ed., Chelsea, New York, 1987. MR 0201608 (34:1490)
  • [9] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. (1907), 191-210.
  • [10] Ch. Pommerenke, Univalent functions, Vandenhoeck and Ruprecht, Göttingen, 1975. MR 0507768 (58:22526)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30C25

Retrieve articles in all journals with MSC: 30C25

Additional Information

Keywords: Univalent functions, maximal covered disks, extremal problems, variational methods, quadratic differentials, logarithmic capacity, transfinite diameter, elliptic integrals
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society