Neighborhoods of univalent functions

TL;DR

This paper proves that univalent functions have neighborhoods consisting solely of univalent functions, establishing stability under small perturbations, and connects this to classical univalence criteria with applications to Taylor series.

Contribution

The paper introduces a new neighborhood stability result for univalent functions and links it to classical univalence criteria, with sharpness analysis and applications.

Findings

Univalent functions are stable under small perturbations.

The main theorem generalizes classical univalence criteria.

The hypothesis of the main result is shown to be sharp.

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.