On Becker's univalence criterion
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CitationHuusko Juha-Matti. Vesikko Toni. (2017). On Becker's univalence criterion. JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, 458 (1) , 781-794. 10.1016/j.jmaa.2017.09.016.
We study locally univalent functions f analytic in the unit disc D of the complex plane such that |f"(z)/f'(z)| (1 − |z|2) ≤ 1 + C(1 − |z|) holds for all z ∈ D, for some C ∈ (0,∞). If C ≤ 1, then f is univalent by Becker's univalence criterion. We discover that for C ∈ (1,∞) 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.