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On some classes of analytic functions. (English) Zbl 0822.30012

In this paper the author improves some results concerning univalent, starlike or convex functions in the unit disc \(E = \{z : | z | < 1\}\) due to R. Singh and S. Singh (1989), Owa (1989, 1991), Nunokawa (1990), Z. Stankiewicz (1990) and Keun Lee and Owa (1991).
In particularly he proved: I. If \(f(z) = z + a_ 2 z^ 2 + \cdots\) satisfies in \(E\) the condition \[ \left | {zf''(z) \over f'(z)} \right | \leq \begin{cases} 3/2 - \alpha \quad & \text{if } \alpha \in [0,1/2] \\ (1 - \alpha) [1 + 1/(2 \alpha)] \quad & \text{if } \alpha \in (1/2, 2/3] \\ (1 - \alpha) (3 - \alpha)/(2 - \alpha) \quad & \quad \alpha \in (2/3,1), \end{cases} \] then \(f \in S^* (\alpha) = \{f : \text{Re} (zf' (z)/f(z)) > \alpha\), \(z \in E\}\).
II. If \(f(z) = z + a_ 2 z^ 2 + \cdots\) satisfies in \(E\) the condition \(\text{Re}\{f''(z) + zf''(z)\}^{1/2} > 1/2\) for \(z \in E\), then \(\text{Re} f' (z) > 1/2\) for \(z \in E\) and \(\text{Re}\{f(z)/z\} > \log 2\) for \(z \in E\).

MSC:

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