A New Class of Meromorphic Functions Involving the Polylogarithm Function

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 A of analytic functions in the open unit disk U = {z : |z| < 1}. By making use of the generalized operator they introduced certain new subclasses of A and investigated many related properties [7]. A year later, same authors again employed the nth order polylogarithm function to define a multiplier transformation on the classA in U [8]. To the best of our knowledge, no research work has discussed the polylogarithm function in conjunctionwithmeromorphic functions. Thus, in this present paper, we redefine the polylogarithm function to be on meromorphic type. Let Σ denote the class of functions of the form


Introduction
Historically, the classical polylogarithm function was invented in 1696, by Leibniz and Bernoulli, as mentioned in [1].For || < 1 and  a natural number with  ≥ 2, 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][4][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 A of analytic functions in the open unit disk U = { : || < 1}.By making use of the generalized operator they introduced certain new subclasses of A 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 A in U [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 Σ denote the class of functions of the form which are analytic in the punctured open unit disk A function () in Σ is said to be meromorphically starlike of order  if and only if for some  (0 ≤  < 1).We denote by Σ * () the class of all meromorphically starlike order .Furthermore, a function () in Σ is said to be meromorphically convex of order  if and only if for some  (0 ≤  < 1).We denote by Σ  () the class of all meromorphically convex order .For functions  ∈ Σ given by (2) and  ∈ Σ given by we define the Hadamard product (or convolution) of  and  by Let Σ  be the class of functions of the form which are analytic and univalent in U * .Liu and Srivastava [9] defined a function ℎ  ( 1 , . . .,   ;  1 , . . .,   ; ) by multiplying the well-known generalized hypergeometric function    with  − as follows: where  1 , . . .,   ;  1 , . . .,   are complex parameters and  ≤  + 1,  ∈ N.
Analogous to Liu and Srivastava work [9] and corresponding to a function Φ  () given by we consider a linear operator Ω  () : Σ → Σ which is defined by the following Hadamard product (or convolution): Next, we define the linear operator   () : Σ → Σ as follows: Now, by making use of the operator   (), we define a new subclass of functions in Σ  as follows.
In the following sections, we investigate coefficient inequalities, extreme points, radii of starlikeness and convexity of order , and integral means inequalities for the new class  , (, ).

Coefficient Inequalities
The following theorem gives a necessary and sufficient condition for a function  to be in the class  , (, ).Theorem 2. Let  ∈ Σ  given by (8).
Proof.Suppose that  ∈  , (, ).Then If we choose  to be real and letting  → 1, we get Hence,  ∈  , (, ).Finally, we note that inequality (14) is sharp; the extremal function is

Extreme Points
In this section, we determine the extreme points for functions in the class  , (, ).
Considering that ∑ ∞ =1 |  | < 1, the above expression is less than 1 −  if and only if By Theorem 2, we have then, (27) holds true if which is equivalent to which yields the starlikeness of the family and completes the proof.

Integral Means Inequalities
Let () and () be analytic in U * .Then the function () is said to be subordinate to () in U * , written by if there exists a function () which is analytic in U * with (0) = 0 and |()| < 1 with  ∈ U * and such that () = (()) for  ∈ U * .From the definition of the subordinations, it is easy to show that subordination (37) implies that Theorem 6 (see [10]).(43) From Theorem 6, it suffices to prove that (47) That completes the proof.