2017/05/16 by Juha-Matti Huusko, Huusko, Juha-Matti, Toni Vesikko +1
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Meromorphic and Entire Functions #math.CV #msc:34C10 #msc:34M10
paper · pdf · doi:10.48550/arxiv.1705.05738
14 pages, 1 figure
arxiv created 2017/05/16 · arxiv updated 2017/05/17
We study locally univalent functions f analytic in the unit disc \mathbbD of the complex plane such that |f"(z)/f'(z)|(1-|z|2)≤ 1+C(1-|z|) holds for all z∈\mathbbD, for some 0<C<∞. If C≤ 1, then f is univalent by Becker's univalence criterion. We discover that for 1<C<∞ 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.