On Becker's univalence criterion

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.