## Some applications of a first order differential subordination.(English)Zbl 1083.30016

Let $$p$$ and $$q$$ be analytic functions in the unit disc $$E={z:| z| <1}$$, with $$p(0)=q(0)=1$$. Assume that $$\alpha$$ and $$\delta$$ are real numbers such that $$0<\delta\leq 1$$, $$\alpha+\delta\geq 0$$. Let $$\beta$$ and $$\gamma$$ be complex numbers with $$\beta\neq 0$$. The authors investigate the differential subordination $p^{\alpha}(z)\left(p(z)+\dfrac{zp'(z)}{\beta p(z)+\gamma}\right)^{\delta}\prec q^{\alpha}(z)\left(q(z)+\dfrac{zq'(z)}{\beta q(z)+\gamma}\right)^{\delta}$ for $$z\in E$$, and obtained several sufficient conditions for starlikeness and univalence of functions analytic in the unit disc $$E$$.

### 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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