Neighborhoods of univalent functions

By Mihai N. Pascu, Nicolae R. Pascu. 2009

176 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1870230d8a41fa2adc3aed8fb2727aaf2cb786af

Citations in the corpus

• Uniformly Convex and Uniformly Starlike Functions (2011, unknown): “In the case when ϕ is a convex univalent function, this result is a special case of the following general result: Theorem 3.5 (Ruscheweyh [55, Theorem 1, p. 275]).”
• A generalized operator involving the $q$-hypergeometric function (2012, unknown): “We observe that M γ (f) ≡ Nγ(f), M(1) γ (f) ≡ Mγ(f), where Nγ(f) is called a γneighborhood of f introduced by Ruscheweyh [23] and Mγ(f) was defined by Silverman [28].”