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Classes of meromorphically multivalent functions associated with the generalized hypergeometric function. (English) Zbl 1049.30008

Making use of a linear operator, which is defined here by means of a Hadamard product (or convolution) involving the generalized hypergeometric function, the authors introduce and investigate two novel classes of meromorphically multivalent functions. In recent years, many important properties and charatcteristics of various interesting subclasses of the class \(\Sigma_{p}\) of meromorphically \(p\)-valent functions were investigated extensively by (among others) Aouf et al. Joshi and Srivastava, Kulkarni et al., Liu and Srivastava, Owa et al., Srivastava et al., Uralegaddi and Somanatha, and Yang. The main object of this paper is to present several inclusion and other properties of functions in the classes \(\Omega_{p, q, s}(\alpha_{1}; A, B)\) and \(\Omega_{p, q,s}^{+}(\alpha_{1}; A, B)\) which the authors introduced here. They also apply the familiar concept of neighborhoods of analytic functions to meromorphically \(p\)-valent functions in the class \(\Sigma_{p}\). In the first results the authors apply the well-known Jack’s Lemma.

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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