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Sufficient conditions for starlikeness. (English) Zbl 0998.30010
Let $$A$$ be the class of functions $$f$$ of the form $$f(z)=z+a_2 z^2+ \dots$$ which are holomorphic in the unit disc $$U=\{z\in \mathbb{C}:|z|<1\}$$. A function $$f\in A$$ is said to be starlike of order $$\alpha$$ if it satisfies $$\text{Re} \{zf'(z)/f(z)\} >\alpha$$ for $$z\in U$$. We denote by $$S^* (\alpha)$$, $$0\leq\alpha <1$$, the class of such functions and $$S^*=S^*(0)$$.
In this paper the authors obtain some sufficient conditions for starlikeness and for starlikeness of order $$\alpha$$ for the functions $$f\in A$$. For example: Theorem 2: If $$f\in A$$ satisfies $\text{Re} \left\{{zf'(z)\over f(z)}\left( \alpha{zf''(z) \over f'(z)}+1\right) \right\}> -{\alpha^2 \over 4}(1-\alpha),\;z\in U$ for some $$\alpha\in \langle 0,2)$$, then $$f\in S^*({\alpha \over 2})$$.

##### MSC:
 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
##### Keywords:
starlike functions of order $$\alpha$$