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NEIGHBOURHOODS OF UNIVALENT FUNCTIONS

Part of: Miscellaneous topics of analysis in the complex domain Geometric function theory

Published online by Cambridge University Press:  14 September 2010

MIHAI N. PASCU  and
NICOLAE R. PASCU
MIHAI N. PASCU*
Affiliation:
Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Str. Iuliu Maniu Nr. 50, Braşov 500091, Romania (email: mihai.pascu@unitbv.ro)
NICOLAE R. PASCU
Affiliation:
Department of Mathematics, Southern Polytechnic State University, 1100 S. Marietta Pkwy, Marietta, GA 30060-2896, USA (email: npascu@spsu.edu)
*
✉For correspondence; e-mail: mihai.pascu@unitbv.ro
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Abstract

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The main result shows that a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighbourhood consisting entirely of univalent functions. For the particular choice of a linear function in the hypothesis of the main theorem, we obtain a corollary which is equivalent to the classical Noshiro–Warschawski–Wolff univalence criterion. We also present an application of the main result in terms of Taylor series, and we show that the hypothesis of our main result is sharp.

Keywords

neighbourhoods of univalent functionsunivalent functionNoshiro–Warschawski–Wolff univalence criterion

MSC classification

Secondary: 30C55: General theory of univalent and multivalent functions 30E10: Approximation in the complex domain
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 2 , April 2011 , pp. 210 - 219
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

The authors kindly acknowledge the support from CNCSIS research grant PNII-IDEI 209.

References

[1]Pommerenke, Ch., Boundary Behaviour of Conformal Maps (Springer, Berlin, 1992).CrossRefGoogle Scholar