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Differential sandwich theorems for some subclasses of analytic functions. (English) Zbl 1091.30019
Summary: Let $$q_1$$ and $$q_2$$ be univalent in $$\Delta:=\{z:|z|<1\}$$ with $$q_1(0)=q_2(0) =1$$. We give some applications of first-order differential subordination and superordination to obtain sufficient conditions for normalized analytic functions $$f$$ with $$f(0)=f'(0)=1$$ to satisfy $$q_1(z)\prec\frac{z^2f'(z)} {f^2(z)}\prec q_2(z)$$.

##### MSC:
 30C80 Maximum principle, Schwarz’s lemma, Lindelöf principle, analogues and generalizations; subordination 30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)