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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. {\bf 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.