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An univalence criterion and the Schwarzian derivative. (English) Zbl 0945.30015

By using the method of Loewner chains, the author obtains the following main results.
Theorem. Let \(f(z)=z +a_2 z^2 +...\) be analytic in \(|z|< 1\), let \(\alpha\) be a complex number, \(\text{Re} \alpha > 0\) and let \(\{f,z\}\) denote the Schwarzian derivative \(\left( {f''(z) \over f'(z)} \right)' - {1 \over 2} \left( {f''(z) \over f'(z)} \right)^2 \). If \(\left|{(1-\mid z \mid ^{2 \alpha})^2 \over 2 \alpha^2 |z|^2} \left( z^2 \{f,z\}+(1+\alpha) {zf"(z) \over f'(z)}\right)\right|\leq\) for all \(|z |<1\), then \(F_\alpha (z) = \left(\alpha \int_0 ^2 u ^{\alpha-1} f' (u)du \right) ^{1/ \alpha}\) is analytic and univalent in \(|z|< 1\).
For \(\alpha =1\) we have the well-known criterion given by Nehari.

MSC:

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