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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 $\alpha_2$ $(0<\alpha_1, \alpha_2\le 1$. Several interesting consequences of the main results are also given. All these results presented here are sharp.

30C45Special classes of univalent and multivalent functions
Full Text: DOI
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