Abstract
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 Srivastava work [9] and corresponding to a function given by we consider a linear operator which is defined by the following Hadamard product (or convolution): Next, we define the linear operator as follows: Now, by making use of the operator , we define a new subclass of functions in as follows.
Definition 1. For and , let denote a subclass of consisting functions of form (2) satisfying the condition that where is given by (12). Furthermore, we say that a function , whenever is of form (8).
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 .
2. Coefficient Inequalities
The following theorem gives a necessary and sufficient condition for a function to be in the class .
Theorem 2. Let given by (8). Then if and only if
Proof. Suppose that . Then
If we choose to be real and letting , we get
which is equivalent to (14). Conversely, let us suppose that assertion (14) holds true.
Then we can write
Hence, . Finally, we note that inequality (14) is sharp; the extremal function is
3. Extreme Points
In this section, we determine the extreme points for functions in the class .
Theorem 3. Let and Then, if and only if it can be represented in the form
Proof. Let , , , . Then, we have
Therefore,
Hence, by Theorem 2, .
Conversely, we suppose that , since
We set
and . Then we have
The results follow.
4. Radii of Meromorphic Starlikeness and Meromorphic Convexity
Theorem 4. Let . Then is meromorphically starlike of order in the disk , where The result is sharp for the extremal function given by (19).
Proof. It is sufficient to show that which easily follows from (4), since Considering that , the above expression is less than if and only if or 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.
Theorem 5. Let . Then is meromorphically convex of order in the disk , where The result is sharp for the extremal function given by
Proof. By using the technique employed in the proof of Theorem 4, we can show that for , and prove that the assertion of the theorem is true.
5. Integral Means Inequalities
Let and be analytic in . Then the function is said to be subordinate to in , written by if there exists a function which is analytic in with and with and such that for . From the definition of the subordinations, it is easy to show that subordination (37) implies that
Theorem 6 (see [10]). If and are any two functions, analytic in , with , then, for and ; ,
Theorem 7. Suppose and is defined by
If there exists an analytic function such that
then, for and ,
Proof. We need to show that From Theorem 6, it suffices to prove that If we set such that we get Clearly, ; then from Theorem 2 we can write That completes the proof.
Conflict of Interests
The authors declare that they have no conflict of interests.
Authors’ Contribution
Both authors read and approved the final paper.
Acknowledgment
The authors would like to thank the center of research and instrumentation (CRIM), National University of Malaysia (UKM), for sponsoring this work under Grant code GUP-2013-004.