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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 $q\left(0\right)=1$ and $q\left(z\right)\ne 0$ $z\in U$. By using the method of differential subordinations, we derive certain conditions involving $q$ and $z{q}^{\text{'}}$ under which the functions $q$ satisfy the following two-sided inequality:

$-\frac{{\alpha }_{2}\pi }{2}<\mathrm{arg}q\left(z\right)<-\frac{{\alpha }_{1}\pi }{2}\phantom{\rule{3.33333pt}{0ex}}\left(z\in U\right)$

for some ${\alpha }_{1}$ and ${\alpha }_{2}$ $\left(0<{\alpha }_{1},{\alpha }_{2}\le 1$. 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
##### References:
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