On certain subclasses of analytic functions associated with hypergeometric functions
type: Item
instance of: scholarly article
title: On certain subclasses of analytic functions associated with hypergeometric functions
zbMATH Open document ID: 1213.30035
author: Gangadharan Murugusundaramoorthy
author: N. Magesh
published in: Applied Mathematics Letters
publication date: 2011-01-20T00:00:00Z
review text: The authors introduce the new class of analytic functions \(TS_p(\lambda,\alpha,\beta)=S_p(\lambda,\alpha,\beta)\cap T\) where \(S_p(\lambda,\alpha,\beta)\) and \(T\) are the subclasses of \[ \mathcal A=\Big\{f \text{ holomorphic in the unit disc } U: f(z)=z+\sum_{n=2}^\infty a_nz^n\Big\} \] defined by the properties \[ \mathrm{Re}\left\{\frac{zf'(z)}{(1-\lambda)f(z)+\lambda zf'(z)}-\alpha\right\}> \beta\left|\frac{zf'(z)}{(1-\lambda)f(z)+\lambda zf'(z)}-1\right|,\quad z\in U, \] and \[ f(z)=z-\sum_{n=2}^\infty |a_n|z^n, \] respectively. As noted in the paper, the class \(TS_p(\lambda,\alpha,\beta)\) is a generalization of many other known classes of analytic functions. The main result gives necessary and sufficient conditions on the parameters \(a,b,c,\lambda, \alpha, \beta \) for functions of the form \(zF(a,b;c;z)\) and \(z[2-F(a,b;c;z)]\) to be in \(TS_p(\lambda,\alpha,\beta)\), where \[ F(a,b;c;z)=z-\left|\frac{ab}{c}\right|\sum_{n=2}^\infty \frac{(a+1)_{n-2}(b+1)_{n-2}}{(c+1)_{n-2}(1)_{n-1}}z^n. \]
reviewed by: Eugen Drăghici
Mathematics Subject Classification ID: 30C45
zbMATH DE Number: 5839402
zbMATH Keywords: starlike function
zbMATH Keywords: convex function
zbMATH Keywords: uniformly starlike function
zbMATH Keywords: uniformly convex function
MaRDI profile type: Publication
full work available at URL: j.aml.2010.10.048
OpenAlex ID: W2019262536
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DOI: 10.1016/J.AML.2010.10.048
description: scientific article; zbMATH DE number 5839402