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volume 6, issue 3, article 68, 2005.

Received 20 October, 2004;

accepted 02 June, 2005.

Communicated by:A. Sofo

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Journal of Inequalities in Pure and Applied Mathematics

NEW SUBCLASSES OF MEROMORPHIC p−VALENT FUNCTIONS

B.A. FRASIN AND G. MURUGUSUNDARAMOORTHY

Department of Mathematics Al al-Bayt University P.O. Box: 130095 Mafraq, Jordan.

EMail:bafrasin@yahoo.com

Vellore Institute of Technology, Deemed University, Vellore, TN-632 014 India.

EMail:gmsmoorthy@yahoo.com

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2000Victoria University ISSN (electronic): 1443-5756 202-04

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New Subclasses of Meromorphicp−Valent

Functions

B.A. Frasin and G. Murugusundaramoorthy

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Abstract

In this paper, we introduce two subclassesΩ^∗_p(α)andΛ^∗_p(α)of meromorphic p-valent functions in the punctured diskD = {z : 0 < |z| < 1}.Coefficient inequalities, distortion theorems, the radii of starlikeness and convexity, closure theorems and Hadamard product ( or convolution) of functions belonging to these classes are obtained.

2000 Mathematics Subject Classification:30C45, 30C50.

Key words: Meromorphicp−valent functions, Meromorphically starlike and convex functions.

1. Introduction and Definitions

LetΣ_p denote the class of functions of the form:

(1.1) f(z) = 1

z^p +

∞

X

n=1

ap+n−1z^p+n−1 (p∈N),

which are analytic andp-valent in the punctured unit diskD = {z : 0 <|z| <

1}.A functionf ∈Σ_pis said to be in the classΩ_p(α)of meromorphicp-valently starlike functions of orderαinDif and only if

(1.2) Re

−zf⁰(z) f(z)

> α (z ∈ D; 0≤α < p; p∈N).

Furthermore, a functionf ∈Σpis said to be in the classΛp(α)of meromorphic p-valently convex functions of orderαinDif and only if

(1.3) Re

−1− zf⁰⁰(z) f⁰(z)

> α (z ∈ D; 0≤α < p; p∈N).

The classesΩ_p(α), Λ_p(α)and various other subclasses ofΣ_phave been stud- ied rather extensively by Aouf et.al. [1] – [3], Joshi and Srivastava [4], Kulkarni et. al. [5], Mogra [6], Owa et. al. [7], Srivastava and Owa [8], Uralegaddi and Somantha [9], and Yang [10].

In the next section we derive sufficient conditions forf(z)to be in the classes Ω_p(α)andΛ_p(α),which are obtained by using coefficient inequalities.

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2. Coefficient Inequalities

Theorem 2.1. Let σ_n(p, k, α) = (p+n+k −1) +|p+n+ 2α−k−1|. If f(z)∈Σp satisfies

(2.1)

∞

X

n=1

σ_n(p, k, α)|a_p+n−1|<2(p−α)

for someα(0≤α < p)and somek(k ≥p),thenf(z)∈Ω_p(α).

Proof. Suppose that (2.1) holds true for α (0 ≤ α < p)and k (k ≥ p). For f(z)∈Σp,it suffices to show that

zf⁰(z) f(z) +k

zf⁰(z)

f(z) + (2α−k)

<1 (z ∈ D).

We note that

zf⁰(z) f(z) +k

zf⁰(z)

f(z) + (2α−k)

=

k−p+P∞

n=1(p+n+k−1)ap+n−1z^2p+n−1 2α−k−p+P∞

n=1(p+n+ 2α−k−1)ap+n−1z^2p+n−1

≤ k−p+P∞

n=1(p+n+k−1)|ap+n−1| |z|^2p+n−1 p+k−2α−P∞

n=1|p+n+ 2α−k−1| |a_p+n−1| |z|^2p+n−1

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< k−p+P∞

n=1(p+n+k−1)|ap+n−1| p+k−2α−P∞

n=1|p+n+ 2α−k−1| |a_p+n−1|. The last expression is bounded above by 1 if

k−p+

∞

X

n=1

(p+n+k−1)|ap+n−1|< p+k−2α−

∞

X

n=1

|p+n+ 2α−k−1| |ap+n−1| which is equivalent to our condition (2.1) of the theorem.

Example 2.1. The functionf(z)given by

(2.2) f(z) = 1 z^p +

∞

X

n=1

4(p−α)

n(n+ 1)σ_n(p, k, α)z^p+n−1 (p∈N) belongs to the classΩ_p(α).

Sincef(z)∈Ω_p(α)if and only ifzf⁰(z)∈Λ_p(α),we can prove:

Theorem 2.2. Iff(z)∈Σp satisfies (2.3)

∞

X

n=1

(p+n−1)σ_n(p, k, α)|ap+n−1|<2(p−α) for someα(0≤α < p)and somek(k ≥p),thenf(z)∈Λp(α).

Example 2.2. The functionf(z)given by

(2.4) f(z) = 1 z^p +

∞

X

n=1

4(p−α)

n(n+ 1)(p+n−1)σ_n(p, k, α)z^p+n−1 belongs to the classΛp(α).

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In view of Theorem2.1and Theorem2.2, we now define the subclasses:

Ω^∗_p(α)⊂Ω_p(α)andΛ^∗_p(α)⊂Λ_p(α),

which consist of functions f(z)∈ Σ_p satisfying the conditions (2.1) and (2.3), respectively.

Letting p = 1, 1 ≤ k ≤ n+ 2α, where 0 ≤ α < 1 in Theorem2.1 and Theorem2.2, we have the following corollaries:

Corollary 2.3. Iff(z)∈Σ₁satisfies

∞

X

n=1

(n+α)|a_n|<1−α

thenf(z) ∈Ω₁(α) = Σ^∗(α)the class of meromorphically starlike functions of orderαinD.

Corollary 2.4. Iff(z)∈Σ₁satisfies

∞

X

n=1

n(n+α)|a_n|<1−α

thenf(z)∈Λ1(α) = Σ^∗_K(α)the class of meromorphically convex functions of orderαinD.

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3. Distortion Theorems

A distortion property for functions in the classΩ^∗_p(α)is contained in

Theorem 3.1. If the function f(z) defined by (1.1) is in the classΩ^∗_p(α), then for0<|z|=r <1,we have

1

r^p − 2(p−α)

p+k+|p+ 2α−k|r^p ≤ |f(z)|

(3.1)

≤ 1

r^p + 2(p−α)

p+k+|p+ 2α−k|r^p, and

p

r^p+1− 2p(p−α)

p+k+|p+ 2α−k|r^p−1 (3.2)

≤ |f⁰(z)|

≤ p

r^p+1 + 2p(p−α)

p+k+|p+ 2α−k|r^p−1.

The bounds in (3.1) and (3.2) are attained for the functionsf(z)given by

(3.3) f(z) = 1

z^p + 2(p−α)

p+k+|p+ 2α−k|z^p (p∈N; z ∈ D).

Proof. Sincef ∈Ω^∗_p(α),from the inequality (2.1), we have (3.4)

∞

X

n=1

|ap+n−1| ≤ 2(p−α)

p+k+|p+ 2α−k|.

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Thus, for0<|z|=r <1,and making use of (3.4) we have

|f(z)| ≤

1 z^p

+

∞

X

n=1

|ap+n−1| |z|^p+n−1 (3.5)

≤ 1 r^p +r^p

∞

X

n=1

|ap+n−1|

≤ 1

r^p + 2(p−α)

p+k+|p+ 2α−k|r^p and

|f(z)| ≥

1 z^p

−

∞

X

n=1

|ap+n−1| |z|^p+n−1 (3.6)

≥ 1 r^p −r^p

∞

X

n=1

|ap+n−1|

≥ 1

r^p − 2(p−α)

p+k+|p+ 2α−k|r^p. We also observe that

(3.7) p+k+|p+ 2α−k|

p

∞

X

n=1

(p+n−1)|ap+n−1| ≤2(p−α)

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which readily yields the following distortion inequalities:

|f⁰(z)| ≤ p

|z|^p+1 +

∞

X

n=1

(p+n−1)|ap+n−1| |z|^p+n−2 (3.8)

≤ p

r^p+1 +r^p−1

∞

X

n=1

(p+n−1)|ap+n−1|

≤ p

r^p+1 + 2p(p−α)

p+k+|p+ 2α−k|r^p−1 and

|f⁰(z)| ≥ p

|z|^p+1 −

∞

X

n=1

(p+n−1)|ap+n−1| |z|^p+n−2 (3.9)

≥ p

r^p+1 −r^p−1

∞

X

n=1

(p+n−1)|a_p+n−1|

≥ p

r^p+1 − 2p(p−α)

p+k+|p+ 2α−k|r^p−1. This completes the proof of Theorem3.1.

Similarly, for functionf(z)∈Λ^∗_p(α),and making use of (2.3), we can prove Theorem 3.2. If the function f(z)defined by (1.1) is in the classΛ^∗_p(α), then

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for0<|z|=r <1,we have 1

r^p − 2(p−α)

p[p+k+|p+ 2α−k|]r^p ≤ |f(z)|

(3.10)

≤ 1

r^p + 2(p−α)

p[p+k+|p+ 2α−k|]r^p, and

p

r^p+1− 2(p−α)

p+k+|p+ 2α−k|r^p−1 (3.11)

≤ |f⁰(z)|

≤ p

r^p+1 + 2(p−α)

p+k+|p+ 2α−k|r^p−1.

The bounds in (3.10) and (3.11) are attained for the functionsf(z)given by

(3.12) g(z) = 1

z^p + 2(p−α)

p[p+k−1 +|p+ 2α−k|]z^p (p∈N; z ∈ D).

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4. Radii of Starlikeness and Convexity

The radii of starlikeness and convexity for the classesΩ^∗_p(α)is given by

Theorem 4.1. If the functionf(z)be defined by (1.1) is in the classΩ^∗_p(α),then f(z)is meromorphicallyp-valently starlike of orderδ(0 ≤δ < p)in|z| < r₁, where

(4.1) r1 = inf

n≥1

(p−δ)σ_n(p, k, α) 2(3p+n+ 1−δ)(p−α)

_2p+n−1¹

(p∈N).

Furthermore,f(z) is meromorphicallyp-valently convex of orderδ(0≤δ < p) in|z|< r₂,where

(4.2) r₂ = inf

n≥1

p(p−δ)σ_n(p, k, α)

2[(p+n−1)[3p+n−1−δ](p−α)

_2p+n−1¹

(p∈N).

The results (4.1) and (4.2) are sharp for the functionf(z)given by

(4.3) f(z) = 1

z^p + 2(p−α)

σ_n(p, k, α)z^p+n−1 (p∈N; z ∈ D).

Proof. It suffices to prove that

(4.4)

zf⁰(z) f(z) +p

≤p−δ,

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for|z| ≤r₁. We have

zf⁰(z) f(z) +p

=

P∞

n=1(2p+n−1)a_p+n−1z^p+n−1

1

z^p +P∞

n=1a_p+n−1z^p+n−1 (4.5)

≤ P∞

n=1(2p+n−1)|ap+n−1| |z|^2p+n−1 1−P∞

n=1|ap+n−1| |z|^2p+n−1 . Hence (4.5) holds true if

(4.6)

∞

X

n=1

(2p+n−1)|ap+n−1| |z|^2p+n−1

≤(p−δ) 1−

∞

X

n=1

|ap+n−1| |z|^2p+n−1

! ,

or (4.7)

∞

X

n=1

3p+n−1−δ

(p−δ) |ap+n−1| |z|^2p+n−1 ≤1, with the aid of (2.1), (4.7) is true if

(4.8) 3p+n−1−δ

(p−δ) |z|^2p+n−1 ≤ σ_n(p, k, α)

2(p−α) (n ≥1).

Solving (4.8) for|z|, we obtain

(p−δ)σ (p, k, α) ¹

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In precisely the same manner, we can find the radius of convexity asserted by (4.2), by requiring that

(4.10)

zf⁰⁰(z)

f⁰(z) +p+ 1

≤p−δ, in view of (2.1). This completes the proof of Theorem4.1.

Similarly, we can get the radii of starlikeness and convexity for functions in the classΛ^∗_p(α).

Theorem 4.2. If the functionf(z)be defined by (1.1) is in the classΛ^∗_p(α),then f(z)is meromorphicallyp-valently starlike of orderδ(0 ≤δ < p)in|z| < r₃, where

(4.11) r₃ = inf

n≥1

(p−δ)(p+n−1)σ_n(p, k, α) 2(3p+n+ 1−δ)(p−α)

_2p+n−1¹

(p∈N).

Furthermore,f(z) is meromorphicallyp-valently convex of orderδ(0≤δ < p) in|z|< r4,where

(4.12) r₄ = inf

n≥1

p(p−δ)(p+n−1)σ_n(p, k, α) 2[(p+n−1)[3p+n−1−δ](p−α)

_2p+n−1¹

(p∈N).

The results (4.11) and (4.12) are sharp for the functiong(z)given by

(4.13) g(z) = 1

z^p + 2(p−α)

(p+n−1)σ_n(p, k, α)z^p+n−1 (p∈N; z ∈ D).

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5. Closure Theorems

Let the functionsf_j(z)be defined, forj ∈ {1,2, . . . , m},by (5.1) f_j(z) = 1

z^p +

∞

X

n=1

ap+n−1,jz^p+n−1, (z ∈ D).

Now, we shall prove the following results for the closure of functions in the classesΩ^∗_p(α)andΛ^∗_p(α).

Theorem 5.1. Let the functionsf_j(z), j ∈ {1,2, . . . , m}, defined by (5.1) be in the classΩ^∗_p(α).Then the functionh(z)∈Ω^∗_p(α)where

(5.2) h(z) =

m

X

j=1

b_j f_j(z), b_j ≥0 and

m

X

j=1

b_j = 1).

Proof. From (5.2), we can writeh(z)as

(5.3) h(z) = 1

z^p +

∞

X

n=1

cp+n−1z^p+n−1,

where

(5.4) cp+n−1 =

m

X

j=1

b_jap+n−1,j, j ∈ {1,2, . . . , m}.

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Since f_j(z)∈Ω^∗_p(α),(j ∈ {1,2, . . . , m}), from (2.1) , we have

∞

X

n=1

σ_n(p, k, α) 2(p−α)

^m X

j=1

b_j|ap+n−1,j|

!

=

m

X

j=1

b_j

∞

X

n=1

σ_n(p, k, α)

2(p−α) |a_p+n−1,j|

!

≤

m

X

j=1

bj = 1,

which shows that h(z)∈Ω^∗_p(α).This completes the proof of Theorem5.1.

Using the same technique as in the proof of Theorem5.1, we have

Theorem 5.2. Let the functionsf_j(z), j ∈ {1,2, . . . , m}, defined by (5.1) be in the classΛ^∗_p(α).Then the functionh(z)∈Λ^∗_p(α),whereh(z)defined by (5.2).

Theorem 5.3. Let

(5.5) fp−1(z) = 1

z^p (z ∈ D) and

(5.6) fp+n−1(z) = 1

z^p + 2(p−α)

σn(p, k, α)z^p+n−1,

where n ∈ N0 = N∪ {0}; z ∈ D.Thenf(z) ∈ Ω^∗_p(α)if and only if it can be expressed in the form

(5.7) f(z) =

∞

X

n=0

λp+n−1fp+n−1(z)

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whereλp+n−1 ≥0,(n ∈N0)andP∞

n=0λp+n−1 = 1.

Proof. From (5.5), (5.6) and (5.7), it is easily seen that

f(z) =

∞

X

n=0

λp+n−1fn+p−1(z) (5.8)

= 1

z^p + 2(p−α)

σ_n(p, k, α)λp+n−1z^p+n−1. Since

∞

X

n=1

σ_n(p, k, α)

2(p−α) . 2(p−α)

σn(p, k, α)λp+n−1 =

∞

X

n=1

λp+n−1 = 1−λp−1 ≤1, it follows from Theorem2.1that the functionf(z)given by (5.6) is in the class Ω^∗_p(α).

Conversely, let us suppose thatf(z)∈Ω^∗_p(α).Since

|ap+n−1| ≤ 2(p−α)

σ_n(p, k, α) (n ≥1), setting

λp+n−1 = σ_n(p, k, α)

2(p−α) |ap+n−1|, (n ≥1)

and ∞

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it follows that

f(z) =

∞

X

n=0

λp+n−1fp+n−1(z).

This completes the proof of the theorem.

Similarly, we can prove the same result for the classΛ^∗_p(α).

Theorem 5.4. Let

(5.9) gp−1(z) = 1

z^p (z ∈ D) and

(5.10) gp+n−1(z) = 1

z^p + 2(p−α)

(p+n−1)σ_n(p, k, α)z^p+n−1

wheren ∈N⁰ andz ∈ D.Theng(z)∈Λ^∗_p(α)if and only if it can be expressed in the form

(5.11) g(z) =

∞

X

n=0

λp+n−1gp+n−1(z)

whereλ_p+n−1 ≥0,(n ∈N0)andP∞

n=0λ_p+n−1 = 1.

Next, we state a theorem which exhibits the fact that the classesΩ^∗(α)and Λ^∗_p(α)are closed under convex linear combinations. The proof is fairly straight- forward so we omit it.

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Theorem 5.5. Suppose thatf(z)andg(z)are in the classΩ^∗(α)(or in Λ^∗_p(α)).

Then the functionh(z)defined by

(5.12) h(z) = tf(z) + (1−t)g(z), (0≤t≤1) is also in the classΩ^∗(α)(or in Λ^∗_p(α)).

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6. Convolution Properties

For functions

(6.1) f_j(z) = 1 z^p +

∞

X

n=1

ap+n−1,jz^p+n−1, (j = 1,2)

belonging to the classΣ_p,we denote by(f₁∗f₂)(z)the Hadamard product (or convolution) of the functionsf₁(z)andf₂(z),that is,

(6.2) (f₁∗f₂)(z) = 1 z^p +

∞

X

n=1

ap+n−1,1ap+n−1,2z^p+n−1.

Finally, we prove the following.

Theorem 6.1. Let each of the functionsfj(z) (j = 1,2)defined by (6.1) be in the classΩ^∗(α).Then(f₁∗f₂)(z)∈Ω^∗(η),where

1

2(k+ 1−p−n)≤η= p([p+k+|p+ 2α−k|]²−4(p−α)²) 4(p−α)²+ [p+k+|p+ 2α−k|]² , (6.3)

(k ≥p; p, n∈N).

The result is sharp.

Proof. Forf_j(z)∈Ω^∗(α) (j = 1,2),we need to find the largestηsuch that (6.4)

∞

X

n=1

σ_n(p, k, η)

2(p−η) |ap+n−1,1| |ap+n−1,2| ≤1.

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From (2.1), we have (6.5)

∞

X

n=1

σ_n(p, k, α)

2(p−α) |ap+n−1,1| ≤1 and

(6.6)

∞

X

n=1

σ_n(p, k, α)

2(p−α) |ap+n−1,2| ≤1.

Therefore, by the Cauchy-Schwarz inequality, we have (6.7)

∞

X

n=1

σ_n(p, k, α) 2(p−α)

q

|ap+n−1,1| |ap+n−1,2| ≤1.

Thus it is sufficient to show that (6.8) σ_n(p, k, η)

2(p−η) |a_p+n−1,1| |a_p+n−1,2|

≤ σn(p, k, α) 2(p−α)

q

|ap+n−1,1| |ap+n−1,2|, (n ≥1) that is, that

(6.9)

q

|ap+n−1,1| |ap+n−1,2| ≤ (p−η)σ_n(p, k, α)

(p−α)σ_n(p, k, η), (n≥1).

From (6.7), we have

2(p−α)

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Consequently, we need only to prove that

(6.10) 2(p−α)

σ_n(p, k, α) ≤ (p−η)σn(p, k, α)

(p−α)σ_n(p, k, η), (n≥1).

Letη ≥ ¹₂ (k+ 1−p−n),wherek ≥ pandp, n ∈N.It follows from (6.10) that

(6.11) η ≤ p[σ_n(p, k, α)]²−4(p−α)²(p+n−1)

4(p−α)²+ [σ_n(p, k, α)]² = Ψ(n).

SinceΨ(k)is an increasing function of n(n ≥ 1),lettingn = 1 in (6.11), we obtain

(6.12) η ≤Ψ(1) = p([p+k+|p+ 2α−k|]²−4(p−α)²) 4(p−α)²+ [p+k+|p+ 2α−k|]² , which proves the main assertion of Theorem6.1.

Finally, by taking the functions (6.13) f_j(z) = 1

z^p + 2(p−α)

σ_n(p, k, α)z^p+n−1, (j = 1,2) we can see the result is sharp.

Similarly, and as the above proof, we can prove the following.

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Theorem 6.2. Let each of the functions f_j(z) (j = 1,2)defined by (6.1) be in the classΛ^∗_p(α).Then(f1∗f2)(z)∈Λ^∗_p(ξ),where

1

2(k+ 1−p−n)≤ξ= p(p[p+k+|p+ 2α−k|]²−4(p−α)²) 4(p−α)²+p[p+k+|p+ 2α−k|]² , (6.14)

(k ≥p; p, n∈N).

The result is sharp for the functions

(6.15) f_j(z) = 1

z^p + 2(p−α)

(p+n−1)σn(p, k, α)z^p+n−1, (j = 1,2).

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References

[1] M.K. AOUF, New criteria for multivalent meromorphic starlike functions of order alpha, Proc. Japan. Acad. Ser. A. Math. Sci., 69 (1993), 66–70.

[2] M.K. AOUFANDH.M. HOSSEN, New criteria for meromorphicp-valent starlike functions, Tsukuba J. Math., 17 (1993) 481–486.

[3] M.K. AOUF AND H.M. SRIVASTAVA, A new criteria for meromorphic p-valent convex functions of order alpha, Math. Sci. Res. Hot-line, 1(8) (1997), 7–12.

[4] S.B. JOSHI AND H.M. SRIVASTAVA, A certain family of meromorphi- cally multivalent functions, Computers Math. Appl., 38 (1999), 201–211.

[5] S.R. KUKARNI, U.H. NAIK AND H.M. SRIVASTAVA, A certain class of meromorphicallyp-valent quasi-convex functions, Pan Amer. Math. J., 8(1) (1998), 57–64.

[6] M.L. MOGRA, Meromorphic multivalent functions with positive coeffi- cients I and II, Math. Japon., 35 (1990), 1–11 and 1089–1098.

[7] S. OWA, H.E. DARWISH AND M.K. AOUF, Meromorphic multivalent functions with positive and fixed second coefficients, Math. Japon., 46 (1997), 231–236.

[8] H.M. SRIVASTAVA AND S. OWA (Eds.), Current Topics in Analytic Function Theory, World Scientific, Singapore/New Jersey/London/Hong Kong, (1992).

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[9] B.A. URALEGADDI AND C. SOMANATHA, Certain classes of mero- morphic multivalent functions, Tamkang J. Math., 23 (1992), 223–231.

[10] D.G. YANG, On new subclasses of meromorphic p-valent functions, J.

Math. Res. Exposition, 15 (1995) 7–13.