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Argument inequalities for certain analytic functions. (English) Zbl 1205.30014

Summary: Let \(q\) be analytic in the open unit disk \(U\) with \(q(0)=1\) and \(q(z)\neq 0\) \(z\in U\). By using the method of differential subordinations, we derive certain conditions involving \(q\) and \(zq'\) under which the functions \(q\) satisfy the following two-sided inequality: \[ -\frac{\alpha_2\pi}{2}<\mathrm{arg} q(z)<-\frac{\alpha_1\pi}{2}~(z\in U) \] for some \(\alpha_1\) and \(\alpha_2\) \((0<\alpha_1, \alpha_2\leq 1\). Several interesting consequences of the main results are also given. All these results presented here are sharp.

MSC:

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

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