A linear operator and associated families of meromorphically multivalent functions of order a

Abstract

Making use of a linear operator, which is defined here by means of the Hadamard product (or convolution), we introduce two novel subclasses $Q_{a,c}(p,\alpha ;A,B)$ and $Q_{a,c}^{+}(p,\alpha ;A,B)$ of meromorphically multivalent functions of order $\alpha$ $(0\leqq \alpha <p)$ in the punctured unit disk $\mathbb{U}^{\ast}$. The main object of the present paper is to investigate the various important properties and characteristics of these subclasses of meromorphically multivalent functions. We extend the familiar concept of neighborhoods of analytic functions to these subclasses of meromorphically multivalent functions. We also derive many interesting results for the Hadamard products of functions belonging to the function class $Q_{a,c}^{+}(p,\alpha;A,B)$.