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 \[ f_{x}(z)=z+2(1-\alpha) \sum_{k=2}^{\infty}\frac{x^{k-1}z^{k}}{k[ 1+(k-1)\lambda]^{n}},\quad | x| =1,\;z\in\Delta\, . \]


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