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)



On univalent functions with two preassigned values

Authors: Maxwell O. Reade and Eligiusz J. Złotkiewicz
Journal: Proc. Amer. Math. Soc. 30 (1971), 539-544
MSC: Primary 30.42
MathSciNet review: 0283189
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\mathfrak{M}_M}$ denote the class of functions analytic and univalent in the unit disc $ \Delta $ subject to the conditions

$\displaystyle f(0) = 0,\quad f({z_0}) = {z_0},\quad \vert f(z)\vert < M,$

where $ {z_0},{z_0} \ne 0$, is a fixed point of $ \Delta $ and $ 1 \leqq M \leqq \infty $.

In the present note, we determine by the method of circular symmetrization, the exact value of the ``Koebe constant'' for the class $ {\mathfrak{M}_M}$. We also determine Koebe sets for the class $ \mathfrak{M}_M^ \ast $ consisting of the starlike functions, and for $ \mathfrak{M}_M^\alpha $, consisting of all functions mapping $ \Delta $ onto domains convex in the direction $ {e^{i\alpha }}$.

By ``Koebe set'' we understand the set $ \mathcal{K}({\mathfrak{M}_M}),\mathcal{K}({\mathfrak{M}_M})$.

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

  • [1] Jan Krzyż and Maxwell O. Reade, Koebe domains for certain classes of analytic functions, J. Analyse Math. 18 (1967), 185–195. MR 0212175,
  • [2] J. Krzyź and E. Złotkiewicz, Koebe sets for univalent functions with two preassigned values, Ann. Acad. Sci. Fenn. Ser. AI (to appear).
  • [3] Zdzisław Lewandowski, Sur certaines classes de fonctions univalentes dans le cercle-unité, Ann. Univ. Mariae Curie-Sklodowska Sect. A 13 (1959), 115–126 (French, with Polish and Russian summaries). MR 0120383
  • [4] G. Pick, Über die konforme Abbildung eines kreises ... , Wien. Berichte 126 (1917), 247-263.
  • [5] Werner Rogosinski, Über den Wertevorrat einer analytischen Funktion, von der zwei Werte vorgegeben sind, Compositio Math. 3 (1936), 199–226 (German). MR 1556940

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 30.42

Retrieve articles in all journals with MSC: 30.42

Additional Information

Keywords: Univalent function, Koebe constant, Koebe domain, starlike mapping, convex in one direction, hyperbolic measure, Green's function, bounded functions
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society