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A generalization of Lewandowski’s univalence criterion. (English) Zbl 0882.30011
Let the function $f$ be analytic in the unit disk $U= \{z:|z|<1\}$, $f(0)= f'(0)-1 =0$ and let $\alpha$ be the complex number, $\text{Re} \alpha >0$. If there exists an analytic function $p$, $\text{Re} p(z) >0$, $z\in U$ such that $$\left|{1-p(z) \over 1+ p(z)} \right ||z|^{2 \text{Re} \alpha} +{1- |z|^{2 \text{Re} \alpha} \over \text{Re} \alpha} \left|{zf''(z) \over f'(z)} +{zp'(z) \over p(z)+1} \right|\le 1, \quad z\in U \backslash \{0\}$$ then for all complex $\beta$, $\text{Re} \beta \ge\text{Re} \alpha$, the function $F_\beta (z)= [\beta \int^z_0 u^{\beta-1} f'(u)du]^{1\over \beta}$ is analytic and univalent in $U$.

MSC:
 30C55 General theory of univalent and multivalent functions