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Argument properties of analytic functions associated with the fixed coefficients. (English) Zbl 1372.30010

Summary: In the present paper, we derive a property of analytic functions \(p(z) = 1 + p_{n}z^{n} + \cdots \) with fixed initial coefficients in their series expansion, which satisfy the condition \( -\pi \beta /2 < \arg p(z_{1}) < \arg p(z) < \arg p(z_{2}) = \pi \alpha /2\), for some \(z_{1}\) and \(z_{2}\) with \(|z_{1}|=|z_{2}|=r<1\) and for all \(z\) with \(|z|<r\), where \(0<\alpha \leq 2\) and \(0<\beta \leq 2\). Using this property, we obtain some sufficient conditions for normalized analytic functions \(f(z) = z + a_{n+1}z^{n+1} + \cdots \) by considering the fixed initial coefficients to satisfy \(-\pi \beta /2 < \arg \left\{ zf'(z)/f(z) - \gamma \right\} < \pi \alpha /2\) for all \(z\) in the unit disk \(\mathbb {U}\) on the complex plane, where \(0 \leq \alpha\), \(\beta < 1\), and \(\gamma =0\) or 1/2.

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
30C50 Coefficient problems for univalent and multivalent functions of one complex variable
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