A New Class of Meromorphic Functions Involving the Polylogarithm Function

We introduce a new operator associated with polylogarithm function. By making use of the new operator, we define a certain new class of meromorphic functions and discussed some important properties of it. 1. Introduction Historically, the classical polylogarithm function was invented in 1696, by Leibniz and Bernoulli, as mentioned in [1]. For and a natural number with , the polylogarithm function (which is also known as Jonquiere’s function) is defined by the absolutely convergent series: Later on, many mathematicians studied the polylogarithm function such as Euler, Spence, Abel, Lobachevsky, Rogers, Ramanujan, and many others [2], where they discovered many functional identities by using polylogarithm function. However, the work employing polylogarithm has been stopped many decades later. During the past four decades, the work using polylogarithm has again been intensified vividly due to its importance in many fields of mathematics, such as complex analysis, algebra, geometry, topology, and mathematical physics (quantum field theory) [3–5]. In 1996, Ponnusamy and Sabapathy discussed the geometric mapping properties of the generalized polylogarithm [6]. Recently, Al-Shaqsi and Darus generalized Ruscheweyh and Salagean operators, using polylogarithm functions on class of analytic functions in the open unit disk . By making use of the generalized operator they introduced certain new subclasses of and investigated many related properties [7]. A year later, same authors again employed the th order polylogarithm function to define a multiplier transformation on the class in [8]. To the best of our knowledge, no research work has discussed the polylogarithm function in conjunction with meromorphic functions. Thus, in this present paper, we redefine the polylogarithm function to be on meromorphic type. Let denote the class of functions of the form which are analytic in the punctured open unit disk A function in is said to be meromorphically starlike of order if and only if for some . We denote by the class of all meromorphically starlike order . Furthermore, a function in is said to be meromorphically convex of order if and only if for some . We denote by the class of all meromorphically convex order . For functions given by (2) and given by we define the Hadamard product (or convolution) of and by Let be the class of functions of the form which are analytic and univalent in . Liu and Srivastava [9] defined a function by multiplying the well-known generalized hypergeometric function with as follows: where are complex parameters and , . Analogous to Liu and