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On univalent functions defined by a generalized Sălăgean operator. (English) Zbl 1072.30009
The author introduces a class of univalent functions $$R^{n}(\lambda ,\alpha )$$ defined by a generalized Salagean differential operator $$D^{n}f(z)$$, $$n\in\mathbb{N}_{0}=\{ 0,1,2,\dots\}$$, where $$D^{0}f(z)=f(z)$$, $$D^{1}f(z)=(1-\lambda )f(z)+\lambda zf^{\prime}(z)=D_{\lambda}f(z)$$, $$\lambda\geq 0$$, and $$D^{n}f(z)=D_{\lambda}( D^{n-1}f(z))$$, through: Let $$R^{n}(\lambda ,\alpha )$$ denote the class of functions $$f\in A$$ which satisfy the condition Re$$( D^{n}f(z)) ^{\prime} >\alpha$$, $$z\in\Delta,$$ for some $$0\leq\alpha\leq 1$$, $$\lambda\geq 0$$, and $$n\in\mathbb{N}_{0}$$. Inclusion relations, extreme points of $$R^{n}(\lambda ,\alpha )$$, some convolution properties of functions belonging to $$R^{n}(\lambda ,\alpha )$$ are given. For example:
Theorem. $$R^{n+1}(\lambda ,\alpha )\subset R^{n}(\lambda ,\alpha ).$$
Theorem. Let $$f\in R^{n+1}(\lambda ,\alpha )$$. Then $$f\in R^{n}(\lambda ,\beta )$$, where $$\beta= {\frac{2\lambda^{2}+(1+3\lambda )\alpha}{(1+\lambda)(1+2\lambda )} }\geq \alpha$$.
Theorem. The extremal points of $$R^{n}(\lambda ,\alpha )$$ are $f_{x}(z)=z+2(1-\alpha) \sum_{k=2}^{\infty}\frac{x^{k-1}z^{k}}{k[ 1+(k-1)\lambda]^{n}},\quad | x| =1,\;z\in\Delta\, .$

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