Mathematical Theory and Modeling                                                                                          ...
Mathematical Theory and Modeling                                                                   www.iiste.orgISSN 2224-...
Mathematical Theory and Modeling                                                                        www.iiste.orgISSN ...
Mathematical Theory and Modeling                                                                      www.iiste.orgISSN 22...
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On a certain family of meromorphic p valent functions

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  1. 1. Mathematical Theory and Modeling www.iiste.orgISSN 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.comAbstractIn this paper, we introduce and study a new subclass ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ of meromorphically p-valent functions withnegative coefficients. We first obtained necessary and sufficient conditions for a certain meromorphicallyp-valent function to be in the class‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ, we then investigated the convex combination of certainmeromorphic functions as well as the distortion theorems and convolution properties.AMS mathematics Subject Classification: 30C45.Key Words: Meromophic, p-valent, Distortion, Convolution.1.IntroductionLet ‫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 ൏ 1ሽan which have a simple pole at the origin ሺ‫ ݖ‬ൌ 0ሻ with residue 1 there. Altintas et all [1] defines the function‫ܯ‬ሺ‫ߚ ,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 byCho et al [2, 3]. While Firas Ghanim and Maslina Darus [4] obtained various properties of the function of theform ଵ݂ሺ‫ݖ‬ሻ ൌ ൅ ∑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: ଵ݂ሺ‫ݖ‬ሻ ൌ െ ∑ஶ ܽ௞ ‫ ݖ‬௞ ሺܽ௞ ൒ 0; ‫ ܰ ∈ ݌‬ൌ ሼ1,2, … ሽሻ ௞ୀ௣ ௭೛Let ‫ܣ‬௪ ሺ‫ݖ‬ሻ denote the set of functions analytic in ‫ ܦ‬given by ଵିఈ݂ሺ‫ݖ‬ሻ ൌ െ ∑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 function݂ሺ‫ݖ‬ሻ 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 classFor some ߚሺߚ ൏ 1ሻ, ߙሺ0 ൑ ߙ ൏ 1ሻand ߛሺ0 ൑ ߛ ൏ 1ሻ, for all ‫.ܦ ∈ ݓ ,ݖ‬The purpose of this paper is to investigate some properties of the functions belonging to the class ‫ܯ‬௣ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ. 57
  2. 2. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 20122.Main ResultsTheorem 1 Let the function ݂ሺ‫ݖ‬ሻ be in the class ‫ܣ‬௪ ሺ‫ݖ‬ሻ. Then ‫ܣ‬௪ ሺ‫ݖ‬ሻ belong to the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ .ݓ‬ሻ if andonly 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 .ݓ‬ሻThen we have from (4) that ଵିఈ ି௣ሺଵିఈሻReሼሺ‫ 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 െ ݊ߚTheorem 2. Let the function ݂ሺ‫ݖ‬ሻ be in ‫ܣ‬௪ ሺ‫ݖ‬ሻ and in the function ݃ሺ‫ݖ‬ሻ be defined by ଵିఈሺ௭ି௪ሻ೛ െ ∑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 ൑ 1proof: 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 െ ߛൌ(1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ ሺ0 ൑ ߙ ൏ 1, ߚ ൏ 1, 0 ൑ ߛ ൏ 1, 0 ൑ ߣ ൑ 1ሻThis complete the proof of Theorem 2. 58
  3. 3. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 20123.Distortion TheoremsTheorem 3.: Let ݂ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,∝ ,ݓ‬ሻ, then ଵିఈ ሺଵିఈሻሺଵା௣ఉሻିఊ | ݂ሺ‫ݖ‬ሻ| ൑ + |‫ ݖ‬െ ‫|ݓ‬௡ (7) |௭ି௪|೛ ଵି௡ఉProof : By Theorem 1,we have ሺଵିఈሻሺଵା௣ఉሻିఊ∑ஶ ܽ௡ ൑ ௡ୀ௣ , ሺ0 ൑ ߙ ൏ 1, ߚ ൏ 1, 0 ൑ ߛ ൏ 1, ݊ ∈ ܰ, ݊ߚ ൏ 1 (8) ଵି௡ఉAnd 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 െ ݊ߚWhich 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 െ ݊ߚThis completes the proof of the Theorem 4.4.Convolution Properties4.Let the convolution of two complex-valued meromorphic functions ଵିఈ݂ଵ ሺ‫ݖ‬ሻ ൌ +∑ஶ ܽ௡ଵ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ ௡ୀଵ ሺ௭ି௪ሻ೛And ଵିఈ݂ଶ 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௣ ௡ୀଵTheorem 5. Let the function ‫ܨ‬ሺ‫ݖ‬ሻ be in the class ‫ܣ‬௪ ሺ‫ݖ‬ሻ. Then ‫ܨ‬ሺ‫ݖ‬ሻ belong to the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ if andonly if ஶ ෍ሺ1 െ ݊ߚሻܽ௡ଵ ܽ௡ଶ ൑ ሺ1 െ ߙሻሺ1 ൅ ‫ߚ݌‬ሻ െ ߛ ௡ୀ௣Proof: supposed that 59
  4. 4. Mathematical Theory and Modeling www.iiste.orgISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)Vol.2, No.8, 2012 ‫ܨ‬ሺ‫ݖ‬ሻ ∈ ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻThen we have from (4) that ଵିఈ ି௣ሺଵିఈሻRe {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 getሺ1 െ 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 that݂ሺ‫ݖ‬ሻ ∈ ‫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 byሺଵିఈሻ +∑∞ b୬ଵ ܾ௡ଶ ሺ‫ ݖ‬െ ‫ݓ‬ሻ௡ , ܾ௡ଵ ܾ௡ଶ ൒ 0 ௡ୀଵሺ௭ି௪ሻ೛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௣ ௡ୀଵIs also in the class ‫ܯ‬௉ ሺ‫ߛ ,ߚ ,ߙ ,ݓ‬ሻ ,where ܿ௡ ൌ ሺ1 െ ߣሻܽ௡ଵ ܽ௡ଶ ൅ ߣܾ௡ଵ ܾ௡ଶ ; 0 ൑ ߣ ൑ 1proof: 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 ൑ 1ሻThis completes the proof of Theorem6.References[1] O., Altintas H. Irmak and H.M. Strivastava A family of meromorphically univalent functions with positivecoefficient. DMS-689-IR 1994.[2] N.E. Cho, S.Owa, S.H. Lee and O. Altintas, Generalization class of certain meromorphic univalentfunctions 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 with 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 positivecoefficients, 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 –valentfunctions with negative coefficients. General Mathematics (2011) 19(1) 109-122 60
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