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NEIGHBOURHOODS OF UNIVALENT FUNCTIONS

Published online by Cambridge University Press:  14 September 2010

MIHAI N. PASCU*
Affiliation:
Faculty of Mathematics and Computer Science, Transilvania University of Braşov, Str. Iuliu Maniu Nr. 50, Braşov 500091, Romania (email: [email protected])
NICOLAE R. PASCU
Affiliation:
Department of Mathematics, Southern Polytechnic State University, 1100 S. Marietta Pkwy, Marietta, GA 30060-2896, USA (email: [email protected])
*
For correspondence; e-mail: [email protected]
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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.

Type
Research Article
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