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)



Support points and double poles

Author: Say Song Goh
Journal: Proc. Amer. Math. Soc. 122 (1994), 463-468
MSC: Primary 30C50; Secondary 30C70
MathSciNet review: 1197537
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper gives some sufficient conditions for support points of the class S of univalent functions to be rotations of the Koebe function $ k(z) = z{(1 - z)^{ - 2}}$. If f is a support point associated with a continuous linear functional L and if the function $ \Phi (w) = L({f^2}/(f - w))$ does not have a double pole, then under some mild additional assumptions, a rational support point f must be a rotation of the Koebe function. The situation is more complicated when $ \Phi $ has a double pole. However, we are able to prove the two-functional conjecture for derivative functionals, where $ \Phi $ has a double pole.

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

  • [1] L. Brickman, Functionals of rational type over the class S, Proc. Amer. Math. Soc. 92 (1984), 372-376. MR 759655 (86a:30031)
  • [2] P. L. Duren, Univalent functions, Grundlehren Math. Wiss., vol. 259, Springer-Verlag, New York, Berlin, 1983. MR 708494 (85j:30034)
  • [3] S. S. Goh, The two-functional conjecture for univalent functions, Ph.D. Thesis, Univ. of Michigan, Ann Arbor, MI, 1992. MR 1284366 (95b:30022)
  • [4] -, Functionals of higher derivative type, in progress.
  • [5] K. Pearce, New support points of S and extreme points of HS, Proc. Amer. Math. Soc. 81 (1981), 425-428. MR 597655 (82b:30012)
  • [6] A. C. Schaeffer and D. C. Spencer, Coefficient regions for schlicht functions, Amer. Math. Soc. Colloq. Publ., vol. 35, Amer. Math. Soc., Providence, RI, 1950. MR 0037908 (12:326c)
  • [7] U. Srebro, Is the slit of a rational slit mapping in S straight?, Proc. Amer. Math. Soc. 96 (1986), 65-66. MR 813811 (87e:30029)

Similar Articles

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

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

Additional Information

Keywords: Univalent functions, variational methods, algebraic functions, the two-functional conjecture, derivative functionals
Article copyright: © Copyright 1994 American Mathematical Society

American Mathematical Society