Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Univalent functions which map onto regions of given transfinite diameter


Authors: P. L. Duren and M. M. Schiffer
Journal: Trans. Amer. Math. Soc. 323 (1991), 413-428
MSC: Primary 30C70; Secondary 30C50, 30C85
DOI: https://doi.org/10.1090/S0002-9947-1991-0979964-2
MathSciNet review: 979964
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: By a variational method, the sharp upper bound is obtained for the second coefficients of normalized univalent functions which map the unit disk onto regions of prescribed transfinite diameter, or logarithmic capacity.


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

  • [1] Bateman Manuscript Project (A. Erdélyi, ed), Higher transcendental functions, Vol. II, McGraw-Hill, New York, 1953.
  • [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] Peter L. Duren, Univalent functions, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 259, Springer-Verlag, New York, 1983. MR 708494
  • [4] 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
  • [5] Einar Hille, Analytic function theory. Vol. II, Introductions to Higher Mathematics, Ginn and Co., Boston, Mass.-New York-Toronto, Ont., 1962. MR 0201608
  • [6] J. A. Hummel, Lagrange multipliers in variational methods for univalent functions, J. Analyse Math. 32 (1977), 222–234. MR 0463426, https://doi.org/10.1007/BF02803581
  • [7] G. Pick, Uber die konforme Abbildung eines Kreises auf ein schlichtes und zugleich beschränktes Gebiet, S.-B. Kaiserl. Akad. Wiss. Wien 126 (1917), 247-263.
  • [8] G. Pólya, Beitrag zur Verallgemeinerung des Verzerrungssatzes auf mehrfach zusammenhängende Gebiete. I, II, S.-B. Preuss. Akad. Wiss. (1928), 228-232; 280-282.
  • [9] G. Pólya and G. Szegà, Aufgaben und Lehrsätze aus der Analysis, Zweiter Band, Springer-Verlag, Berlin, 1925; English transl. Problems and theorems in analysis, vol. II, Springer-Verlag, New York, 1976.
  • [10] M. Schiffer, A method of variation within the family of simple functions, Proc. London. Math. Soc. 44 (1938), 432-449.
  • [11] Menahem Schiffer, Hadamard’s formula and variation of domain-functions, Amer. J. Math. 68 (1946), 417–448. MR 0018750, https://doi.org/10.2307/2371824
  • [12] Menahem Schiffer and Donald C. Spencer, Functionals of finite Riemann surfaces, Princeton University Press, Princeton, N. J., 1954. MR 0065652
  • [13] G. Szegö, Bemerkungen zu einer Arbeit von Herrn M. Fekete: Über die Verteilung der Wurzeln bei gewissen algebraischen Gleichungen mit ganzzahligen Koeffizienten, Math. Z. 21 (1924), no. 1, 203–208 (German). MR 1544698, https://doi.org/10.1007/BF01187465
  • [14] M. Tsuji, Potential theory in modern function theory, Maruzen Co., Ltd., Tokyo, 1959. MR 0114894

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30C70, 30C50, 30C85

Retrieve articles in all journals with MSC: 30C70, 30C50, 30C85


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1991-0979964-2
Keywords: Univalent functions, second coefficient, extremal problems, variational methods, quadratic differentials, transfinite diameter, logarithmic capacity, elliptic integrals
Article copyright: © Copyright 1991 American Mathematical Society