# A New Class of Meromorphic Functions Involving the Polylogarithm Function

Authors: Khadeejah Rasheed Alhindi; Maslina Darus
Journal: Journal of Complex Analysis
 (2014)

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## Abstract

We introduce a new operatorDcf(z) 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.


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## Body

## 1. Introduction

Historically, the classical polylogarithm function was invented in 1696, by Leibniz and Bernoulli, as mentioned in [1]. For |z|<1 and c a natural number with c≥2, the polylogarithm function (which is also known as Jonquiere’s function) is defined by the absolutely convergent series:
(1)Lic(z)=∑k=1∞‍zkkc.
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 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.LetΣ denote the class of functions of the form
(2)f(z)=1z+∑k=0∞‍akzk,
which are analytic in the punctured open unit disk
(3)U*≔{z:z∈C,0<|z|<1}≕U∖{0}.
A function f(z) in Σ is said to be meromorphically starlike of order δ if and only if
(4)R{-zf′(z)f(z)}>δ;(z∈U*),
for some δ(0≤δ<1). We denote by Σ*(δ) the class of all meromorphically starlike order δ. Furthermore, a function f(z) in Σ is said to be meromorphically convex of order δ if and only if
(5)R{-1-zf′′(z)f′(z)}>δ;(z∈U*),
for some δ(0≤δ<1). We denote by ΣK(δ) the class of all meromorphically convex order δ. For functions f∈Σ given by (2) and g∈Σ given by
(6)g(z)=1z+∑k=0∞‍bkzk,
we define the Hadamard product (or convolution) of f and g by
(7)(f*g)(z)=1z+∑k=0∞‍akbkzk.
Let Σp be the class of functions of the form
(8)f(z)=1z+∑k=0∞‍akzk;ak≥0,
which are analytic and univalent in U*.Liu and Srivastava [9] defined a function hp(α1,…,αq;β1,…,βs;z) by multiplying the well-known generalized hypergeometric function qFs with z-p as follows:
(9)hp(α1,…,αq;β1,…,βs;z)=z-pqFs(α1,…,αq;β1,…,βs;z),
where α1,…,αq;β1,…,βs are complex parameters and q≤s+1, p∈N.Analogous to Liu and Srivastava work [9] and corresponding to a function Φc(z) given by
(10)Φc(z)=z-2Lic(z)=1z+∑k=0∞‍1(k+2)czk,
we consider a linear operator Ωcf(z):Σ→Σ which is defined by the following Hadamard product (or convolution):
(11)Ωcf(z)=Φc(z)*f(z)=1z+∑k=0∞‍1(k+2)cakzk.
Next, we define the linear operator Dcf(z):Σ→Σ as follows:
(12)Dcf(z)={Ωcf(z)-12ca0}=1z+∑k=1∞‍1(k+2)cakzk.
Now, by making use of the operator Dcf(z), we define a new subclass of functions in Σp as follows.Definition 1.
Forα>1 and 0<β≤1, let Nc(α,β) denote a subclass of Σ consisting functions of form (2) satisfying the condition that
(13)R{zDcf(z)-αz2(Dcf(z))′}>β;(z∈U*),
where Dcf(z) is given by (12). Furthermore, we say that a function f∈Nc,p(α,β), whenever f(z) is of form (8).In the following sections, we investigate coefficient inequalities, extreme points, radii of starlikeness and convexity of orderδ, and integral means inequalities for the new class Nc,p(α,β).


## 2. Coefficient Inequalities

The following theorem gives a necessary and sufficient condition for a functionf to be in the class Nc,p(α,β).Theorem 2.
Letf∈Σp given by (8). Then f∈Nc,p(α,β) if and only if
(14)∑k=1∞‍(αk-1)(k+2)cak≤1+α-β.Proof.
Suppose thatf∈Nc,p(α,β). Then
(15)R{z(1z+∑k=1∞‍1(k+2)cakzk)-αz2(-1z2+∑k=1∞‍k(k+2)cakzk-1)}=R{1+α-∑k=1∞‍αk-1(k+2)cakzk+1}>β.
If we choosez to be real and letting z→1, we get
(16)1+α-∑k=1∞‍(αk-1)(k+2)cak>β,
which is equivalent to (14). Conversely, let us suppose that assertion (14) holds true.
Then we can write(17)|zDcf(z)-αz2(Dcf(z))′|=|1+α-∑k=1∞‍αk-1(k+2)cakzk+1|≥1+α-|∑k=1∞‍αk-1(k+2)cakzk+1|>β.
Hence, f∈Nc,p(α,β). Finally, we note that inequality (14) is sharp; the extremal function is
(18)f(z)=1z+(1+α-β)(3)c(α-1)z;α>1.


## 3. Extreme Points

In this section, we determine the extreme points for functions in the classNc,p(α,β).Theorem 3.
Letf0(z)=1/z and
(19)fk(z)=1z+(1+α-β)(k+2)c(αk-1)zk;(k=1,2,…).
Then, f∈Nc,p(α,β) if and only if it can be represented in the form
(20)f(z)=∑k=0∞‍λkfk(z);(λk≥0,∑k=0∞‍λk=1).Proof.
Letf(z)=∑k=0∞‍λkfk(z), λk≥0, k=0,1,2,…, ∑k=0∞‍λk=1. Then, we have
(21)f(z)=∑k=0∞‍λkfk(z)=λ0f0(z)+∑k=1∞‍λkfk(z)=1z+∑k=1∞‍{λk(1+α-β)(k+2)c(αk-1)}zk.
Therefore,
(22)∑k=1∞‍(αk-1)(k+2)c(1+α-β)(k+2)c(αk-1)λk=(1+α-β)∑k=1∞‍λk=(1+α-β)(1-λ0)≤(1+α-β).
Hence, by Theorem 2, f∈Nc,p(α,β).
Conversely, we suppose thatf∈Nc,p(α,β), since
(23)ak≤(1+α-β)(k+2)c(αk-1);k≥1.
We set
(24)λk=(αk-1)(1+α-β)(k+2)cak;k≥1,
and λ0=1-∑k=1∞‍λk. Then we have
(25)f(z)=∑k=0∞‍λkfk(z)=λ0f0(z)+∑k=1∞‍λkfk(z).
The results follow.


## 4. Radii of Meromorphic Starlikeness and Meromorphic Convexity

Theorem 4.
Letf∈Nc,p(α,β). Then f is meromorphically starlike of order δ(0≤δ<1) in the disk |z|<r1, where
(26)r1=inf⁡k[(1-δk+2-δ)(αk-1)(1+α-β)(k+2)c];(k≥1).
The result is sharp for the extremal function f(z) given by (19).Proof.
It is sufficient to show that(27)|zf′(z)f(z)+1|<1-δ,
which easily follows from (4), since
(28)|zf′(z)f(z)+1|=|∑k=1∞(k+1)akzk+11+∑k=1∞akzk+1|≤∑k=1∞(k+1)|ak||z|k+11-∑k=1∞|ak||z|k+1.
Considering that ∑k=1∞‍|ak|<1, the above expression is less than 1-δ if and only if
(29)∑k=1∞‍(k+1)|ak||z|k+1≤(1-δ)(1-∑k=1∞|ak||z|k+1),
or
(30)∑k=1∞‍k+2-δ1-δ|ak||z|k+1≤1.
By Theorem 2, we have
(31)∑k=1∞‍(αk-1)(k+2)c(1+α-β)ak≤1;
then, (27) holds true if
(32)k+2-δ1-δ|z|k+1<(αk-1)(k+2)c(1+α-β);k≥1,
which is equivalent to
(33)|z|k+1<1-δk+2-δ(αk-1)(1+α-β)(k+2)c,
which yields the starlikeness of the family and completes the proof.Theorem 5.
Letf∈Nc,p(α,β). Then f is meromorphically convex of order δ(0≤δ<1) in the disk |z|<r2, where
(34)r2=inf⁡k[(1-δk+2-δ)(αk-1)k(1+α-β)(k+2)c];(k≥1).
The result is sharp for the extremal function f(z) given by
(35)fk(z)=1z+k(1+α-β)(k+2)c(αk-1)zk;(k=1,2,…).Proof.
By using the technique employed in the proof of Theorem4, we can show that
(36)|zf′′(z)f′(z)+2|<1-δ,
for |z|<r2, and prove that the assertion of the theorem is true.


## 5. Integral Means Inequalities

Letf(z) and g(z) be analytic in U*. Then the function f(z) is said to be subordinate to g(z) in U*, written by
(37)f(z)≺g(z);(z∈U*),
if there exists a function w(z) which is analytic in U* with w(0)=0 and |w(z)|<1 with z∈U* and such that f(z)=g(w(z)) for z∈U*. From the definition of the subordinations, it is easy to show that subordination (37) implies that
(38)f(0)=g(0),f(U*)⊂g(U*).Theorem 6 (see [10]).
Iff and g are any two functions, analytic in U, with f≺g, then, for η>0 and z=reiθ; (0<r<1),
(39)∫02π‍|f(z)|ηdθ≤∫02π‍|g(z)|ηdθ.Theorem 7.
Supposef∈Nc,p(α,β) and fk is defined by
(40)fk(z)=1z+(1+α-β)(k+2)c(αk-1)zk;(k=1,2,…).
If there exists an analytic function w(z) such that
(41)[w(z)]k+1=(αk-1)(1+α-β)(k+2)c∑k=1∞‍akzk+1,
then, forz=reiθ and (0<r<1),
(42)∫02π‍|f(reiθ)|ηdθ≤∫02π‍|fk(reiθ)|ηdθ;(η>0).Proof.
We need to show that(43)∫02π‍|1+∑k=1∞‍akzk+1|ηdθ≤∫02π‍|1+(1+α-β)(k+2)c(αk-1)zk+1|ηdθ.
From Theorem 6, it suffices to prove that
(44)1+∑k=1∞‍akzk+1≺1+(1+α-β)(k+2)c(αk-1)zk+1.
If we set w such that
(45)[w(z)]k+1=(αk-1)(1+α-β)(k+2)c∑k=1∞‍akzk+1,
we get
(46)1+∑k=1∞‍akzk+1=1+(1+α-β)(k+2)c(αk-1)[w(z)]k+1.
Clearly, w(0)=0; then from Theorem 2 we can write
(47)|w(z)|k+1=|(αk-1)(1+α-β)(k+2)c∑k=1∞‍akzk+1|≤(αk-1)(1+α-β)(k+2)c∑k=1∞‍|ak||z|k+1<|z|<1.
That completes the proof.



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