1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.8, 2012
On a Certain Family of Meromorphic p- valent Functions
with Negative Coefficients
D.O. Makinde
Department of Mathematics,
Obafemi Awolowo University, Ile-Ife 220005, Nigeria.
dmakinde@oauife.edu.ng, domakinde.comp@gmail.com
Abstract
In this paper, we introduce and study a new subclass ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ of meromorphically p-valent functions with
negative coefficients. We first obtained necessary and sufficient conditions for a certain meromorphically
p-valent function to be in the class‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ, we then investigated the convex combination of certain
meromorphic functions as well as the distortion theorems and convolution properties.
AMS mathematics Subject Classification: 30C45.
Key Words: Meromophic, p-valent, Distortion, Convolution.
1.Introduction
Let ‫ܯ‬௉ denote the class of functions ݂ሺ‫ݖ‬ሻ of the form
ஶ
૚
ࢌሺࢠሻ ൌ ൅ ෍ ܽ௡ ‫ ݖ‬௡ ሺܽ௡ ൒ 0, ‫ ܰ ∈ ݌‬ൌ ሼ1,2,3, … ሽ ሺ1ሻ
ࢠ
௡ୀ௣
Which are analytic and univalent in the punctured unit disk
‫ ܦ‬ൌ ሼ‫ 0 ݀݊ܽ ܥ ∈ ݖ ∶ ݖ‬൏ ല‫ݖ‬ല ൏ 1ሽ
an which have a simple pole at the origin ሺ‫ ݖ‬ൌ 0ሻ with residue 1 there. Altintas et all [1] defines the function
‫ܯ‬ሺ‫ߚ ,ߙ ,݌‬ሻ as the function ݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௣ satisfying the inequality:
Re ሼ‫݂ݖ‬ሺ‫ݖ‬ሻ െ ߙ‫ ݖ‬ଶ ݂ ᇱ ሺ‫ݖ‬ሻሽ ൐ ߚ (2)
For some ߙሺߙ ൐ 1ሻ and ߚሺ0 ൑ ߚ ൏ 1ሻ, for all ‫ .ܦ ∈ ݖ‬Other subclasses of the class ‫ܯ‬௣ were studied recently by
Cho et al [2, 3]. While Firas Ghanim and Maslina Darus [4] obtained various properties of the function of the
form
ଵ
݂ሺ‫ݖ‬ሻ ൌ ൅ ∑ஶ ܽ௡ ‫ ݖ‬௡
௡ୀଵ
௭ି௣
With fixed second coefficient. Also, M.Kamali et al [5] studied various properties of meromorphic
P-valent function of the form:
ଵ
݂ሺ‫ݖ‬ሻ ൌ െ ∑ஶ ܽ௞ ‫ ݖ‬௞ ሺܽ௞ ൒ 0; ‫ ܰ ∈ ݌‬ൌ ሼ1,2, … ሽሻ
௞ୀ௣
௭೛
Let ‫ܣ‬௪ ሺ‫ݖ‬ሻ denote the set of functions analytic in ‫ ܦ‬given by
ଵିఈ
݂ሺ‫ݖ‬ሻ ൌ െ ∑ஶ ܽ௡ሺ௭ି௪ሻ೙ ሺܽ௡ ൒ 0; ‫ ܰ ∈ ݌‬ൌ ሼ1,2, … ሽሻ
௞ୀ௣ (3)
ሺ௭ି௪ሻ೛
Which have a pole of order ‫ ݌‬at ሺ‫ ݖ‬ൌ ‫ݓ‬ሻ, ‫ ܦ ∈ ݖ‬and ‫ ݓ‬is an arbitrary fixed point in ‫ .ܦ‬We define the function
݂ሺ‫ݖ‬ሻ in ‫ܣ‬௪ ሺ‫ݖ‬ሻ to be in the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ if it satisfies the inequality
ܴ݁ሼሺ‫ ݖ‬െ ‫ݓ‬ሻ݂ሺ‫ݖ‬ሻ െ ߚሺ‫ ݖ‬െ ‫ݓ‬ሻଶ ݂′ሺ‫ݖ‬ሻሽ ൐ ߛ (4)
and ‫ ݓ‬is an arbitrary fixed point in ‫ .ܦ‬We define the function ݂ሺ‫ݖ‬ሻ in ܽ௪ ሺ‫ݖ‬ሻ to be in the class
For some ߚሺߚ ൏ 1ሻ, ߙሺ0 ൑ ߙ ൏ 1ሻand ߛሺ0 ൑ ߛ ൏ 1ሻ, for all ‫.ܦ ∈ ݓ ,ݖ‬
The purpose of this paper is to investigate some properties of the functions belonging to the class ‫ܯ‬௣ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ.
57

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2.Main Results
Theorem 1 Let the function ݂ሺ‫ݖ‬ሻ be in the class ‫ܣ‬௪ ሺ‫ݖ‬ሻ. Then ‫ܣ‬௪ ሺ‫ݖ‬ሻ belong to the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ if and
only if
∑ஶ ሺ1 െ ݊ߚሻܽ௡ ൑ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ
௡ୀଵ (5)
Proof: let ݂ሺ‫ݖ‬ሻ be as in (3), suppose that
݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ
Then we have from (4) that
ଵିఈ ି௣ሺଵିఈሻ
Reሼሺ‫ ݖ‬െ ‫ݓ‬ሻ ቀሺ௭ି௪ሻ೛ െ ∑ஶ ܽ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ ቁ െ ߚሺ‫ ݖ‬െ ‫ݓ‬ሻଶ ሾቀሺ௭ି௪ሻ೛శభ െ ∑ஶ ݊ܽ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ିଵ ቁሿሽ
௡ୀ௣ ௡ୀ௣
=Reሼ1 െ ߙ െ ∑ஶ ܽ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ା௣ ൅ ‫ߚ݌‬ሺ1 െ ߙሻ ൅ ∑ஶ ߚ݊ܽ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ାଵ ሽ
௡ୀ௣ ௡ୀ௣
= Reሼሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ∑ஶ ሺ1 െ ݊ߚሻܽ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ା௣ሽ ൐ ߛ ሺ‫ܦ ∈ ݖ‬ሻ, w an arbitrary fixed point in D
௡ୀ௣
If we choose ሺ‫ ݖ‬െ ‫ݓ‬ሻ to be real and let ሺ‫ ݖ‬െ ‫ݓ‬ሻ → 1ି , we get
ஶ
ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ෍ሺ1 െ ݊ߚሻܽ௡ ൑ ߛሺߚ ൏ 1; ൑ ߛ ൏ 1, ߚ݊ ൏ 1ሻ
௡ୀ௣
Which is equivalent to (5)
Conversely, suppose that the inequality (5) holds. Then we have:
|ሺ‫ ݖ‬െ ‫ݓ‬ሻ݂ሺ‫ݖ‬ሻ െ ߚሺ‫ ݖ‬െ ‫ݓ‬ሻଶ ݂ ᇱ ሺ‫ݖ‬ሻ െ ሺ1െ∝ሻሺ1 ൅ ‫ߚ݌‬ሻ|
=|-∑ஶ ሺ1 െ ݊ߚሻܽ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ା௣ | ൑ ∑ஶ ሺ1 െ ݊ߚሻܽ௡ |ሺ‫ ݖ‬െ ‫ݓ‬ሻ|௡ା௣
௡ୀଵ ௡ୀ௣
൑ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ,
ሺ‫ ݓ ,ܦ ∈ ,ݖ‬an arbitrary fixed point in ‫ ߚ ܦ‬൏ 1, ܱ ൑ ߙ ൏ 1,0 ൑ ߛ ൏ 1ሻ which implies that ݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௣ ሺ‫ߛ ,ߚ ,∝ ,ݓ‬ሻ.
This completes the prove of the Theorem 1.
Corollary 1: Let ݂ሺ‫ݖ‬ሻ be in ‫ܣ‬௪ ሺ‫ݖ‬ሻ . If ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,∝ ,ݓ‬ሻ , then
ሺ1 െ ߙሻሺ1 ൅ ‫ ߚ݌‬െ ߛሻ
ܽ௡ ൑ , ݊ ൒ 1, ߚ ൏ 1, ݊ߚ ൏ 1
1 െ ݊ߚ
Theorem 2. Let the function ݂ሺ‫ݖ‬ሻ be in ‫ܣ‬௪ ሺ‫ݖ‬ሻ and in the function ݃ሺ‫ݖ‬ሻ be defined by
ଵିఈ
ሺ௭ି௪ሻ೛
െ ∑ஶ ܾ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ , ܾ௡ ൒ 0
௡ୀଵ (6)
Be in the same class ‫ܯ‬௣ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ. Then the function ݄ሺ‫ݖ‬ሻ defined by
ஶ
1െߙ
݄ሺ‫ݖ‬ሻ ൌ ሺ1 െ ߣሻ݂ሺ‫ݖ‬ሻ ൅ ߣ݃ሺ‫ݖ‬ሻ ൌ െ ෍ ܿ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡
ሺ‫ ݖ‬െ ‫ݓ‬ሻ௣
௡ୀ௣
is also in the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,∝ ,ݓ‬ሻ, where ܿ௡ ൌ ሺ1 െ ߣሻܽ௡ ൅ ߣܾ௡ ; 0 ൑ ߣ ൑ 1
proof: Suppose that each of ݂ሺ‫ݖ‬ሻ, ݃ሺ‫ݖ‬ሻ is in the class ‫ܯ‬௣ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ. Then by (5) we have
ஶ ஶ
෍ሺ1 െ ݊ߚሻܿ௡ ൌ ෍ሺ1 െ ݊ߚሻሾሺ1 െ ߣሻܽ௡ ൅ ܾ௡ ሿ
௡ୀ௣ ௡ୀ௣
ஶ ஶ
ൌ ሺ1 െ ߣሻ ෍ሺ1 െ ݊ߚሻܽ௡ ൅ ߣ ෍ሺ1 െ ݊ߚሻܾ௡
௡ୀ௣ ௡ୀଵ
൑ ሺ1 െ ߣሻ(1 െ ߙሻ ሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ ൅ ߣሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ
ൌ(1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ ሺ0 ൑ ߙ ൏ 1, ߚ ൏ 1, 0 ൑ ߛ ൏ 1, 0 ൑ ߣ ൑ 1ሻ
This complete the proof of Theorem 2.
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3.Distortion Theorems
Theorem 3.: Let ݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,∝ ,ݓ‬ሻ, then
ଵିఈ ሺଵିఈሻሺଵା௣ఉሻିఊ
| ݂ሺ‫ݖ‬ሻ| ൑ + |‫ ݖ‬െ ‫|ݓ‬௡ (7)
|௭ି௪|೛ ଵି௡ఉ
Proof : By Theorem 1,we have
ሺଵିఈሻሺଵା௣ఉሻିఊ
∑ஶ ܽ௡ ൑
௡ୀ௣ , ሺ0 ൑ ߙ ൏ 1, ߚ ൏ 1, 0 ൑ ߛ ൏ 1, ݊ ∈ ܰ, ݊ߚ ൏ 1 (8)
ଵି௡ఉ
And
ሺଵିఈሻሺଵା௣ఉሻିఊ
∑ஶ ݊ܽ௡ ൑ ݊
௡ୀ௣ , ሺ0 ൑ ߙ ൏ 1, ߚ ൏ 1, 0 ൑ ߛ ൏ 1, ݊ ∈ ܰ, ݊ߚ ൏ 1 (9)
ଵି௡ఉ
Let ݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ, we have
1െߙ
∞
݂ሺ‫ݖ‬ሻ ൑ ൅ |‫ ݖ‬െ ‫|ݓ‬௡ ෍ ܽ௡
|‫ ݖ‬െ ‫|ݓ‬௣
௡ୀଵ
1െߙ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ
൑ ൅ |‫ ݖ‬െ ‫|ݓ‬௡
|‫ ݖ‬െ ‫|ݓ‬௣ 1 െ ݊ߚ
Which completes the proof of the Theorem 3.
Theorem 4: Let ݂ሺ‫ݖ‬ሻ ∈ ‫ܣ‬௪ ሺ‫ݖ‬ሻ. If ݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ then
1െߙ ݊ሾሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛሿ
ห݂ ᇱሺ௭ሻ ห ൑ ൅ |‫ ݖ‬െ ‫|ݓ‬௡ିଵ
|‫ ݖ‬െ ‫|ݓ‬௣ାଵ 1 െ ݊ߚ
‫01 ݋݊ ݏ݅ ݁ݎ݄݁ݐ‬
ଵିఈ
ห݂ ᇱሺ௭ሻ ห ൑ |௭ି௪|೛శభ+|‫ ݖ‬െ ‫|ݓ‬௡ିଵ ∑ஶ ݊ܽ௡
௡ୀ௣
1െߙ ݊ሾሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛሿ
൑ ൅ |‫ ݖ‬െ ‫|ݓ‬௡ିଵ
|‫ ݖ‬െ ‫|ݓ‬ ௣ାଵ 1 െ ݊ߚ
This completes the proof of the Theorem 4.
4.Convolution Properties
4.
Let the convolution of two complex-valued meromorphic functions
ଵିఈ
݂ଵ ሺ‫ݖ‬ሻ ൌ +∑ஶ ܽ௡ଵ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡
௡ୀଵ
ሺ௭ି௪ሻ೛
And
ଵିఈ
݂ଶ ሺ‫ݖ‬ሻ ൌ +∑ஶ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡
௡ୀଵ
ሺ௭ି௪ሻ೛
Be defined by
ஶ
1െߙ
‫ܨ‬ሺ‫ݖ‬ሻ ൌ ൫݂ଵ ሺ‫ݖ‬ሻ ∗ ݂ଶ ሺ‫ݖ‬ሻ൯ ൌ ൅ ෍ ܽ௡ଵ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡
ሺ‫ ݖ‬െ ‫ݓ‬ሻ௣
௡ୀଵ
Theorem 5. Let the function ‫ܨ‬ሺ‫ݖ‬ሻ be in the class ‫ܣ‬௪ ሺ‫ݖ‬ሻ. Then ‫ܨ‬ሺ‫ݖ‬ሻ belong to the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ if and
only if
ஶ
෍ሺ1 െ ݊ߚሻܽ௡ଵ ܽ௡ଶ ൑ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ
௡ୀ௣
Proof: supposed that
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‫ܨ‬ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ
Then we have from (4) that
ଵିఈ ି௣ሺଵିఈሻ
Re {ሺ‫ ݖ‬െ ‫ݓ‬ሻ ሺ௭ି௪ሻ೛ െ ∑ஶ ܽ௡ଵ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ ሻ െ ߚሺ‫ ݖ‬െ ‫ݓ‬ሻଶ ሾቀሺ௭ି௪ሻ೛శభ െ ∑ஶ ܽ௡ଵ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ିଵ ቁሿሽ
௡ୀଵ ௡ୀଵ
=Re ሼ1 െ ߙ െ ∑ஶ ܽ௡ଵ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ା௣ ൅ ‫ߚ݌‬ሺ1 െ ߙሻ ൅ ∑ஶ ߚܽ௡ଵ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ାଵ ሽ
௡ୀଵ ௡ୀଵ
=Re {ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ∑ஶ ሺ1 െ ݊ߚሻܽ௡ଵ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ା௣ ሽ ൐ ߛ ሺ‫ܦ ∈ ݖ‬ሻ, ‫ ݓ‬an arbitrary fixed point in ‫ܦ‬
௡ୀ௣
If we choose ሺ‫ ݖ‬െ ‫ݓ‬ሻ to be real and let ሺ‫ ݖ‬െ ‫ݓ‬ሻ → 1ି , we get
ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ∑ஶ ሺ1 െ ݊ߚሻܽ௡ଵ ܽ௡ଶ ൑ ߛ ሺߚ ൏ 1; 0 ൑ ߛ ൏ 1, ߚ݊ ൏ 1ሻ
௡ୀ௣
which is equivalent to(5)
Conversely,suppose that the inequality (5) holds. Then we have:
|ሺ ‫ ݖ‬െ ‫ݓ‬ሻ ݂ሺ‫ݖ‬ሻ- ߚሺ‫ ݖ‬െ ‫ݓ‬ሻଶ ݂ ᇱ ሺ‫ݖ‬ሻ െ ሺ1െ∝ሻሺ1 ൅ ‫ߚ݌‬ሻ|
=|-∑ஶ ሺ1 െ ݊ߚሻܽ௡ଵ ܽ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ା௣ | ൑ ∑ஶ ሺ1 െ ݊ߚሻܽ௡ଵ ܽ௡ଶ |ሺ‫ ݖ‬െ ‫ݓ‬ሻ|௡ା௣
௡ୀଵ ௡ୀ௣
൑ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ,
ሺ‫ܦ ∈ ݖ‬ሻ, ‫ ݓ‬an arbitrary fixed point in ‫ ߚ ܦ‬൏ 1,0 ൑ ߙ ൏ 1,0 ൑ ߛ ൏ 1ሻ which implies that
݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ.
This completes the prove of Theorem 5.
Theorem 6. Let the function ‫ܨ‬ሺ‫ݖ‬ሻ be in‫ܣ‬௪ ሺ‫ݖ‬ሻ and the function ‫ܩ‬ሺ‫ݖ‬ሻ defined by
ሺଵିఈሻ
+∑∞ b୬ଵ ܾ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ , ܾ௡ଵ ܾ௡ଶ ൒ 0
௡ୀଵ
ሺ௭ି௪ሻ೛
Be in the same class ‫ܯ‬௣ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ.Then the function ‫ܪ‬ሺ‫ݖ‬ሻ define by
ሺ1 െ ߙሻ
∞
‫ܪ‬ሺ‫ݖ‬ሻ ൌ ሺ1 െ ߣሻ‫ܨ‬ሺ‫ݖ‬ሻ ൅ ߣ‫ܩ‬ሺ‫ݖ‬ሻ ൌ ൅ ෍ ܿ௡ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡
ሺ‫ ݖ‬െ ‫ݓ‬ሻ௣
௡ୀଵ
Is also in the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ ,where ܿ௡ ൌ ሺ1 െ ߣሻܽ௡ଵ ܽ௡ଶ ൅ ߣܾ௡ଵ ܾ௡ଶ ; 0 ൑ ߣ ൑ 1
proof: suppose that each of ‫ܨ‬ሺ‫ݖ‬ሻ, ‫ܩ‬ሺ‫ݖ‬ሻ is in the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ. Then by 5 we have
∑∞ ሺ1 െ ݊ߚሻܿ௡ = ∑∞ ሺ1 െ ݊ߚሻ ሺ1 െ ߣሻܽ௡ଵ ܽ௡ଶ ൅ ߣܾ௡ଵ ܾ௡ଶ
௡ୀଵ ௡ୀଵ
∞ ∞
ൌ ሺ1 െ ߣሻ ෍ሺ1 െ ݊ߚሻ ܽ௡ଵ ܽ௡ଶ ൅ ߣ ෍ሺ1 െ ݊ߚሻb୬ଵ ܾ௡ଶ
௡ୀଵ ௡ୀଵ
൑ ሺ1 െ ߣሻ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ + ߣሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ
ൌ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛሺ0 ൑ ߙ ൏ 1, ߚ ൏ 1, 0 ൑ ߛ ൏ 1, 0 ൑ ߣ ൑ 1ሻ
This completes the proof of Theorem6.
References
[1] O., Altintas H. Irmak and H.M. Strivastava A family of meromorphically univalent functions with positive
coefficient. DMS-689-IR 1994.
[2] N.E. Cho, S.Owa, S.H. Lee and O. Altintas, Generalization class of certain meromorphic univalent
functions eith positive coefficient, Kyungypook Math, J.(1989) 29, 133-139.
[3] N.E. Cho, , S.H. Lee, S.Owa, A class of meromorphic univalent functions wit
h positive coefficients, Kobe J. Math. (1987) 4, 43-50
[4] F. Ghanim and M. Darus On a certain subclass of meromorphic univalent functions wih fixed second positive
coefficients, surveys in mathematics and applications (2010) 5 49-60.
[5] M. Kamali, K. Suchithra, B.A. Stephen and A. Gangadharan, On certain subclass of meromorphic p –valent
functions with negative coefficients. General Mathematics (2011) 19(1) 109-122
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