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Subclasses of meromorphically multivalent functions associated with a certain linear operator. (English) Zbl 1049.30009

The authors investigate various inclusion and other properties of a certain class of meromorphically p-valent functions which are defined here by means of a linear operator. For a function \(f\) in the class \(\Sigma_{p}\) (analytic and \(p\)-valent in the punctured unit disk) is defined the linear operator \(D^{n}\) by \(D^{0}f(z)=f(z)\), \(D^{1}f(z)={\frac{\left( z^{p+1} f(z) \right) ^{\prime}}{z^{p}}}\), and \(D^{n}f(z)=D(D^{n-1}f(z))\). The linear operator \(D^{n}\) was considered, when \(p=1\), by Uralegaddi and Somanatha. Recently, Aouf and Hossen presented several results involving the operator \(D^{n}\) for \(p\in \mathbb{N}\). Many important properties and characteristics of various interesting subclasses of the class \(\Sigma_{p}\) were investigated by (among others) Aouf, Chen, Cho, Joshi, Liu, Kulkarni, Mogra, Owa, Somanatha, Srivastava, Uralegaddi and Yang. The main object of this paper is to present several inclusion and other properties of functions in the classes \(R_{n,p}(A,B)\) and \(S_{n,p}(A,B)\) which the authors have introduced here. They also extend and apply the familiar concept of neighborhood of analytic functions to meromorphically \(p\)-valent functions in the class \(\Sigma_{p}\).

MSC:

30C45 Special classes of univalent and multivalent functions of one complex variable (starlike, convex, bounded rotation, etc.)
33C20 Generalized hypergeometric series, \({}_pF_q\)
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