Neighborhoods of univalent functions

2009/10/28 by Mihai N. Pascu, Pascu, Mihai N., Nicolae R. Pascu +1
Mathematics · #30C45 #30C55 #30E10 #Complex Variables (math.CV) #FOS: Mathematics #math.CV #msc:30C45 #msc:30C55 #msc:30E10

paper · pdf · doi:10.48550/arxiv.0910.5456

9 pages, 1 figure

arxiv created 2009/10/28 · arxiv updated 2009/12/01

Abstract

The main result shows a small perturbation of a univalent function is again a univalent function, hence a univalent function has a neighborhood 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.

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