A Certain Subclass of Meromorphically Multivalent Analytic Functions with Negative Coefficients

Corresponding Author: Rabha W. Ibrahim Faculty of Computer Science and IT, University Malaya, 50603 Kuala Lumpur, Malaysia Email: rabhaibrahim@yahoo.com Abstract: The paper aims to introduce a certain subclass S * g(A,B,α,p,j) of the class Σ * (p) of meromorphically multivalent functions with negative coefficients, defined by using the definitions of Hadamard product and subordination for two functions belong to the class Σ * (p). We first investigate the geometric characterization property, giving the coefficient estimates for functions in the class S * g(A,B,α,p,j). We also obtain the distortion theorem, radii of meromorphically p-valent starlikeness and convexity of order (0≤γ<p), neighborhood property, partial sums, convolution properties as well as integral operator and integral representation.


Introduction
Let Σ(p) denote the class of functions f(z) of the form: Which are analytic and p-valent (multivalent) in the punctured unit disk U * .
A function f∈Σ * (p) is said to be meromorphically pvalent starlike of order (0≤γ<p) in U * (r) if , also a function f∈Σ * (p) is said to be meromorphically p-valent e.g., Duren, 1983;Goodman, 1983;Srivastava and Owa, 1992). Let B be a subclass of the class Σ * (p). We define the radius of meromorphically p-valent starlike of order γ and the radius of meromorphically p-valent convex of order γ for the class B The convolution or the Hadamard product of two meromorphic p-valent functions f and g, where f is given by (1.1) and

( )
Recall (Aouf and El-Ashwah, 2009) that an analytic function f is said to be subordinate to an analytic function g written f<g, if ( ) ( ) ( ), 1 f z g w z z = < for some analytic function w with |w(z)|<1.
By making use of the following definition for Hadamard product and subordination, a new subclass if it satisfies the following subordination condition: where, -1≤A<B≤1; 0<B≤1; 0≤α<p+j; j≤p≤n,; p∈N; j∈2NU{0} = {0, 2, 4, …}z∈U * . or, equivalently, if: The aim intended to be achieved in the current analysis is to identify coefficient estimates, distortion theorem, radii of meromorphically p-valent star likeness and convexity of order (0≤γ<p), neighborhood property, partial sums. Moreover, the convolution properties and integral operator and integral representation are investigated.

Upper Bounds
In the following section, we establisha characterization property which provides a necessary, sufficient condition for a function f(z), defined by (1.1), belongs to the class * ( , , , , ) g A B p j α S and obtain the coefficient estimates.
Theorem 1. Let the function f(z) be given by (1.1). Then f(z) ∈ * ( , , , , ) if and only if: Proof. Suppose that the function f(z) defined by (1.1) is in the class * ( , , , , ) The choice of z to be real and letting z→1 − through real value when |Re{z}|≤|z| for any z, then the inequality (2.2) directly gives the desired condition in (2.1). Conversely, assume that the condition (2.1) holds true and let |z| = 1, then we have: The result is sharp for the function f(z) given by:

Distortion Theorem
The following theorem proves the distortion inequality for the function property for another class is investigated by many researchers among them Aouf and El-Ashwah (2009).
Theorem 2: If a function f(z) given by (1.1) is in the class * ( , , , , ) g A B p j α S . Then: The result is sharp for the functions f given by: and from Theorem 1 together with: And similarly: The sharpness of the inequality in (3.1) satisfies by the function f(z) given by (3.2).

Radii of Starlikeness and Convexity
This section considers the radii of meromorphically p-valent starlikeness of order γ(0≤γ<p) and meromorphically p-valent convexity of order γ(0≤γ<p) for the functions that belong to the class * ( , , , , ) , by using methods applied by Kamali et al. (2011) and others.
Theorem 3: Let . Then: f is meromorphically p-valent convex of order γ(0≤γ<p) in |z|<r 2 , where: Each of these results is sharp for the function f(z) given by (2.5).
Proof. (i) From the definition (1.1), we obtain: Thus, we have the desired inequality: So, by Theorem 1, the condition (4.3) will be true if: Putting |z| = r 1 in (4.4), we get the radius of starlikeness.
In order to prove the second assertion of Theorem 3, we find from the definition (1.1) that: Hence, by Theorem 1, the condition (4.5) will be satisfied if: Setting |z| = r 1 in (4.6), we obtain the radius of convexity, which completes the proof of Theorem 3.

Neighborhood Property
Depending on the earlier works by (Goodman, 1957;Ruscheweyh, 1981;Liu and Srivastava, 2004;Aouf, 2009;Aouf and El-Ashwah, 2009) that based upon the familiar concept of neighborhood of analytic functions, we introduce the definition of the δ-neighborhood of a function

Proof. It is easily obvious that
if and only if: For any complex number ε with |ε| =, we have: This is contradiction by |σ|<δ and however, we

Partial Sums
By following (Silvia, 1985;Silverman, 1997) Then we have: The results in (6.2) and (6.3) are sharp for n. Proof. We can see from (6.1) that 1< d n <d n +1 (n∈N). Since {d n } is an increasing sequence, we obtain: This proves (6.2). If we take ( )

Convolution Properties
This section concentrates on a way to derive the convolution properties by using Schild and Silverman (1975) techniques. At the beginning, let's recall the following definition: Let the functions f i (z)(i = 1,2) be defined by: The modified Hadamard product (or convolution) of f 1 (z) and f 2 (z) is defined by: The result is sharp for the functions f i (z) (i = 1,2) given by: Proof. In view of Theorem 1, it suffices to prove that: , we have: Therefore, by the Cauchy-Schwarz inequality, we obtain: Hence, we need to show that Or, equivalently, that: From the equality (7.4), we have ; 2 0 n n n p j p j B A p a a n j B n p p j B A b n p p N j α δ α So, it is sufficient to prove that: It is follows from (7.5) that: We easily see that ϕ(n) is an increasing function of n. So we have: Which completes the proof of Theorem 6. Theorem 7: Let the function f 1 (z) defined by (7.1) be in the class * ( , , , , ) g A B p j ε S and the function f 2 (z) defined by (7.1) be in the class The result is sharp for the functions f i (z)(i = 1,2) given b: Theorem 8: Let the functions f i (z)(i = 1,2) defined by (7.1) be in the class * ( , , , , ) g A B p j ε S and b n ≥b p ≥1(n≥p).
The result is sharp for the functions f i (z)(i = 1,2) given by (7.3).

Integral Representation
In the following theorem, we determine the integral representation of functions that belong to the class