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.


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