Proceedings of the American Mathematical Society

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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
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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).


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  • [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] Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and physicists, Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen mit besonderer Berücksichtigung der Anwendungsgebiete. Bd LXVII, Springer-Verlag, Berlin-Göttingen-Heidelberg, 1954. MR 0060642
  • [3] Robin Cunningham, Univalent functions of given transfinite diameter: a maximum modulus problem, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993), no. 2, 249–271. MR 1234733
  • [4] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • [5] P. L. Duren and M. M. Schiffer, Univalent functions which map onto regions of given transfinite diameter, Trans. Amer. Math. Soc. 323 (1991), no. 1, 413–428. MR 979964, 10.1090/S0002-9947-1991-0979964-2
  • [6] G. M. Goluzin, Geometric theory of functions of a complex variable, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR 0247039
  • [7] I. Gradsteyn and I. Ryzhik, Table of integrals, series and products, Academic Press, New York, 1965; enlarged ed., 1980.
  • [8] Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
  • [9] P. Koebe, Über die Uniformisierung beliebiger analytischer Kurven, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. (1907), 191-210.
  • [10] Christian Pommerenke, Univalent functions, Vandenhoeck & Ruprecht, Göttingen, 1975. With a chapter on quadratic differentials by Gerd Jensen; Studia Mathematica/Mathematische Lehrbücher, Band XXV. MR 0507768

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1993-1161400-4
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