On Becker's univalence criterion

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:458

Article

Huusko, Juha-Matti¹  Vesikko, Toni¹ 

[1] Univ Eastern Finland, POB 111, FI-80101 Joensuu, Finland

关键词: Univalence criterion;    Bounded function;    Bloch space;    Normal function;   

DOI  :  10.1016/j.jmaa.2017.09.016

来源: Elsevier

【 摘 要 】

We study locally univalent functions f analytic in the unit disc D of the complex plane such that |f(z)/f'(z)I (1- |z|(2)) <= 1+C(1 |z|) holds for all z is an element of D, for some C is an element of (0, infinity). If C <= 1, then f is univalent by Becker's univalence criterion. We discover that for C is an element of (1, infinity) the function f remains to be univalent in certain horodiscs. Sufficient conditions which imply that f is bounded, belongs to the Bloch space or belongs to the class of normal functions, are discussed. Moreover, we consider generalizations for locally univalent harmonic functions. (C) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free