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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 \(f\in S^*({\alpha \over 2})\).

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)