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An improvement of Becker’s univalence criterion. (English) Zbl 0666.30012

Proc. Commem. Sess. Simion Stoilow, Brasov/Rom. 1987, 43-48 (1988).
[For the entire collection see Zbl 0658.00006.]
Suppose that \(\alpha\) is a complex number with Re \(\alpha\) \(>0\) and that \(f(z)=z+..\). is analytic in the open unit disk U. The author proves that if \[ | \frac{1-| z|^{2\alpha}}{\alpha}\frac{zf''(z)}{f'(z)}| \leq 1\quad for\quad all\quad z\in U, \] then the function \(F_{\alpha}(z)\) is analytic and univalent in U, where \[ F_{\alpha}(z)=[\alpha \int^{z}_{0}u^{\alpha -1} f'(u)du]^{1/\alpha}. \] Similarly, if \[ \frac{1-| z|^{2 Re \alpha}}{Re \alpha}| \frac{zf''(z)}{f'(z)}| \leq 1\quad for\quad all\quad z\in U, \] then the function \(F_{\beta}(z)\) is analytic and univalent in U for every complex number \(\beta\) with Re \(\beta\geq Re \alpha\). As a corollary the author obtains that if \[ (1-| z|^ 2)| \frac{zf''(z)}{f'(z)}| \leq 1\quad for\quad all\quad z\in U, \] then the function \(F_{\beta}(z)\) is analytic and univalent in U for every complex number \(\beta\) with Re \(\beta\geq 1\). This is an extension of a univalence criterion due to J. Becker [J. Reine Angew. Math. 255, 23-43 (1972; Zbl 0239.30015)].
Reviewer: Renate McLaughlin

MSC:

30C55 General theory of univalent and multivalent functions of one complex variable