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Zbl 1072.30009
Al-Oboudi, F.M.
On univalent functions defined by a generalized Sălăgean operator.
(English)
[J] Int. J. Math. Math. Sci. 2004, No. 25-28, 1429-1436 (2004). ISSN 0161-1712; ISSN 1687-0425/e

The author introduces a class of univalent functions $R^{n}(\lambda ,\alpha )$ defined by a generalized Salagean differential operator $D^{n}f(z)$, $n\in\Bbb{N}_{0}=\{ 0,1,2,\dots\}$, where $D^{0}f(z)=f(z)$, $D^{1}f(z)=(1-\lambda )f(z)+\lambda zf^{\prime}(z)=D_{\lambda}f(z)$, $\lambda\ge 0$, 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 $0\le\alpha\le 1$, $\lambda\ge 0$, and $n\in\Bbb{N}_{0}$. 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 $\beta= {\frac{2\lambda^{2}+(1+3\lambda )\alpha}{(1+\lambda)(1+2\lambda )} }\ge \alpha$. Theorem. The extremal points of $R^{n}(\lambda ,\alpha )$ are $$f_{x}(z)=z+2(1-\alpha) \sum_{k=2}^{\infty}\frac{x^{k-1}z^{k}}{k[ 1+(k-1)\lambda]^{n}},\quad \vert x\vert =1,\ z\in\Delta\, .$$
[Dorin Blezu (Sibiu)]
MSC 2000:
*30C45 Special classes of univalent and multivalent functions

Keywords: generalized Salagean operator; convolution properties; inclusion relations

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