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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 $f\left(z\right)=z+{a}_{2}{z}^{2}+···$ be analytic in $|z|<1$, let $\alpha$ be a complex number, $\text{Re}\alpha >0$ and let $\left\{f,z\right\}$ denote the Schwarzian derivative ${\left(\frac{{f}^{\text{'}\text{'}}\left(z\right)}{{f}^{\text{'}}\left(z\right)}\right)}^{\text{'}}-\frac{1}{2}{\left(\frac{{f}^{\text{'}\text{'}}\left(z\right)}{{f}^{\text{'}}\left(z\right)}\right)}^{2}$. If $\left|\frac{{\left(1-\mid z{\mid }^{2\alpha }\right)}^{2}}{2{\alpha }^{2}{|z|}^{2}}\left({z}^{2}\left\{f,z\right\}+\left(1+\alpha \right)\frac{zf"\left(z\right)}{{f}^{\text{'}}\left(z\right)}\right)\right|\le$ for all $|z|<1$, then ${F}_{\alpha }\left(z\right)={\left(\alpha {\int }_{0}^{2}{u}^{\alpha -1}{f}^{\text{'}}\left(u\right)du\right)}^{1/\alpha }$ 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
##### Keywords:
univalent functions; Schwarzian derivative