On univalent functions defined by a generalized Sălăgean operator
type: Item
instance of: scholarly article
title: On univalent functions defined by a generalized Sălăgean operator
zbMATH Open document ID: 1072.30009
DOI: 10.1155/S0161171204108090
published in: International Journal of Mathematics and Mathematical Sciences
publication date: 2005-05-18T00:00:00Z
full work available at URL: https://eudml.org/doc/51673
review text: 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 \[ f_{x}(z)=z+2(1-\alpha) \sum_{k=2}^{\infty}\frac{x^{k-1}z^{k}}{k[ 1+(k-1)\lambda]^{n}},\quad | x| =1,\;z\in\Delta\, . \]
reviewed by: Dorin Blezu
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 2168694
zbMATH Keywords: generalized Salagean operator
zbMATH Keywords: convolution properties
zbMATH Keywords: inclusion relations
author: F. M. Al-Oboudi
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2168694