International Journal of Analysis and Applications
ISSN 2291-8639
Volume 2, Number 1 (2013), 1-18
http://www.etamaths.com

FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS ASSOCIATED

WITH THE DZIOK-RAINA OPERATOR

G. MURUGUSUNDARAMOORTHY1,∗ AND M. K. AOUF2

Abstract. Making use a linear operator, which is defined here by means of the Hadamard product

(or convolution), involving the Wright’s generalized hypergeometric function , we introduce two novel

subclasses
∑

p (q,s,α1; A,B,λ) and
∑+

p (q,s,α1;A,B,λ) of meromorphically multivalent functions of

order λ (0 ≤ λ < p) in the punctured disc U∗. In this paper we investigate the various important
properties and characteristics of these subclasses of meromorphically multivalent functions. We extend
the familiar concept of neighborhoods of analytic functions to these subclasses of meromorphically

multivalent functions . We also derive many interesting results for the Hadamard products of functions

belonging to the class
∑+

p (q,s,α1; A,B,λ).

1. . Introduction

Let
∑
p denote the class of functions of the form :

(1.1) f(z) = z−p +

∞∑
k=1

akz
k−p (p ∈ N = {1, 2, ...}) ,

which are analytic and p-valent in the punctured disc

U∗ = {z : z ∈ C and 0 < |z| < 1} = U\{0}.
For functions f(z) ∈

∑
p given by (1.1), and g(z) ∈

∑
p given by

(1.2) g(z) = z−p +

∞∑
k=1

bkz
k−p (p ∈ N) ,

we define the Hadamard product (or convolution) of f(z) and g(z) by

(1.3) (f ∗g)(z) = z−p +
∞∑
k=1

akbkz
k−p = (g ∗f)(z).

If f(z) and g(z) are analytic in U, we say that f(z) is subordinate to g(z) ; written symbolically as
follows :

f ≺ g or f(z) ≺ g(z) (z ∈ U),
if there exists a Schwarz function w(z) in U such thatf(z) = g(w(z)) (z ∈ U).

Let α1,A1, ...,αq,Aq and β1,B1, ...,βs,Bs (q,s ∈ N) be positive real parameters such that

2010 Mathematics Subject Classification. 30C45.
Key words and phrases. Wright’s generalized hypergeometric function, Hadamard product, meromorphic functions,

neighborhoods.

c©2013 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

1



2 MURUGUSUNDARAMOORTHY AND AOUF

(1.4) 1 +

s∑
n=1

Bn −
q∑

n=1

An ≥ 0.

The Wright generalized hypergeometric function [36] ( see also [33] )

qΨs[(α1,A1), ..., (αq,Aq); (β1,B1), ...., (βs,Bs); z]

= qΨs[(αn,An)1,q ; (βn,Bn)1,s ; z]

is defined by

qΨs[(αn,An)1,q ; (βn,Bn)1,s ; z] =

(1.5)

∞∑
k=0

{
q∏

n=1

Γ(αn + kAn)

}{
s∏

n=1

Γ(βn + kBn)

}−1
zk

k!
(z ∈ U).

If An = 1(n = 1, ...,q) and Bn = 1(n = 1, ..,s), we have the relationship :

(1.6) Ω qΨs[(αn,1)1,q ; (βn,1)1,s ; z] = qFs(α1, ...,αq; β1, ...,βs; z),

where qFs(α1, ...,αq ; β1, ...,βs ; z) is the generalized hypergeometric function ( see for details [9] , [10] ,
[11] , and [18] ) and

(1.7) Ω = (

q∏
n=1

Γ(αn))
−1(

s∏
n=1

Γ(βn)).

The Wright generalized hypergeometric functions were invoked in the geometric function theory ( see [8]
, [25] , [26], [27] and [28] ).
We define a function qφs[(αn,An)1,q ; (βn,Bn)1,s; z ] = Ωz

−p
qΨs[(αn,An)1,q ;

(βn,Bn)1,s ; z] and consider the following linear operator

WHp[(αn,An)1,q ; (βn,Bn)1,s] :
∑

p
→
∑

p
,

defined by the convolution

(1.8) WHp[(αn,An)1,q; (βn,Bn) 1,s]f(z) = qφs[(αn,An)1,q ; (βn,Bn)1,s ; z] ∗f(z),
so that , for a function f(z) of the form (1.1) , we have

(1.9) WHp[(αn,An)1,q; (βn,Bn) 1,s]f(z) = z
−p +

∞∑
k=1

Ωσk(α1)akz
k−p,

where

(1.10) σk(α1) =
Γ(α1 + kA1).....Γ(αq + kAq)

Γ(β1 + kB1).....Γ(βs + kBs)k!
.

If , for convenience , we write

∆p,q,s[α1]f(z) = WHp[(α1,A1), ....., (αq,Aq); (β1,B1), ...., (βs,Bs)]f(z),

then one can easily verify from the definition (1.9) that

(1.11) zA1(∆p,q,s[α1]f(z))
′

= α1∆p,q.s[α1 + 1]f(z) − (α1 + pA1)∆p,q,s[α1]f(z).
For An = 1(n = 1, ...,q) and Bn = 1(n = 1, ...,s), the operator ∆p,q,s[α1]f(z) = Hp,q,s(α1)f(z) was
introduced and studied by Liu and Srivastava [21].



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 3

For fixed parameters A , B and λ(−1 ≤ B < A ≤ 1 ; 0 ≤ λ < p ; p ∈ N), we say that a function
f(z) ∈ Σp is in the class

∑
p(q,s,α1; A,B,λ) of meromorphically p- valent functions in U if it also

satisfies the following subordination condition :

(1.12) α1
(∆p,q,s[α1 + 1]f(z))

′

(∆p,q,s[α1]f(z))
′ − (α1 + pA1) ≺−A1

p + [pB + (A−B)(p−λ)]z
1 + Bz

,

or , by using (1.11) , if it satisfies the following subordination condition :

(1.13)
1

(p−λ)
(1 +

z(∆p,q,s[α1]f(z))
′′

(∆p,q,s[α1]f(z))
′ + λ) ≺−

1 + Az

1 + Bz
(z ∈ U)

or , equivalently , if the following inequality holds true :

(1.14)

∣∣∣∣∣∣∣∣∣
1 +

z(∆p,q,s[α1]f(z))
′′

(∆p,q,s[α1]f(z))
′ + p

B(1 +
z(∆p,q,s[α1]f(z))

′′

(∆p,q,s[α1]f(z))
′ + p) + (A−B)(p−λ)

∣∣∣∣∣∣∣∣∣
< 1 (z ∈ U).

Furthermore, we say that a function f(z) ∈
∑+
p (q,s,α1; A,B,λ) wherever f(z) is of the form (cf. Equation

(1.1)] :

(1.15) f(z) = z−p +

∞∑
k=p

|ak|zk (p ∈ N) .

We note that for An = 1(n = 1, ...,q) and Bn = 1(n = 1, ...,s) the classes
∑
p(q,s,α1; A,B, 0) =

Ωp,q,s(α1; A,B) and
∑+
p (q,s,α1; A,B, 0) = Ω

+
p,q,s(α1; A,B) are studied by Liu and Srivastava [21]. Also

we note that

∑+
p

(q,s,α1; β,−β,λ) =
∑+

p
(q,s,α1; λ,β)

=


f(z) ∈ Σp and

∣∣∣∣∣∣∣∣∣
1 +

z(∆p,q,s[α1]f(z))
′′

(∆p,q,s[α1]f(z))
′ + p

1 +
z(∆p,q,s[α1]f(z))

′′

(∆p,q,s[α1]f(z))
′ −p + 2λ

∣∣∣∣∣∣∣∣∣
< β

(1.16) (z ∈ U; 0 ≤ λ < p; 0 < β ≤ 1; p ∈ N) } .

Meromorphic multivalent functions have been extensively studied by (for example) (Mogra [22] and
[23]), Uralegaddi and Ganigi [34], Uralegaddi and Somanatha [35], Aouf ([4] and [5]), Aouf and Hossen
[6], Srivastava et al. [32], Owa et al. [24], Joshi and Aouf [14], Joshi and Srivastava [15], Aouf et al. [7],
Raina and Srivastava [29], Yang ([37] and [38]), Kulkarni et al. [16], Liu [17] and Liu and Srivastava ([19]
and [20]).

In this paper we investigate the various important properties and characteristics of the classes
∑
p(q,s,α1; A,B,λ)

and
∑+
p (q,s,α1; A,B,λ). Following the recent investigations by Altintas et al. [3, p. 1668], we ex-

tend the concept of neighborhoods of analytic functions, which was considered earlier by (for example)
Goodman [12] and Ruscheweyh [30], to meromorphically multivalent functions, belonging to the classes∑
p(q,s,α1; A,B,λ) and

∑+
p (q,s,α1; A,B,λ). We also derive many interesting results for the Hadamard

products of functions belonging to the p-valently meromorphic function class
∑+
p (q,s,α1; A,B,λ).



4 MURUGUSUNDARAMOORTHY AND AOUF

2. . Inclusion properties of the class
∑
p(q,s,α1; A,B,λ)

For proving our first inclusion result , we shall make use of the following lemma.

Lemma 1. ( see Jack [13] ). Let the (nonconstant) function w(z) be analytic in U with w(0) = 0. If
|w(z)| attains its maximum value on the circle |z| = r < 1 at a point z0 ∈ U, then

(2.1) z0w
′
(z0) = γw(z0) ,

where γ is a real number and γ ≥ 1.

Theorem 1. . Let α1 ∈ R\{0}. If

(2.2) α1 ≥
A1(A−B)(p−λ)

1 + B
(−1 < B < A ≤ 1 ; 0 ≤ λ < p; p ∈ N; A1 > 0) ,

then ∑
p
(q,s,α1 + 1; A,B,λ) ⊂

∑
p

(q,s,α1; A,B,λ) .

Proof. Let f(z) ∈
∑
p(q,s,α1 + 1; A,B,λ) and suppose that

(2.3)
(∆p,q,s[α1 + 1]f(z))

′

(∆p,q,s[α1]f(z))
′ = 1 +

A1(B −A)(p−λ) w(z)
α1(1 + Bw(z))

,

where the function w(z) is either analytic or meromorphic in U, with w(0) = 0. By differeniating (2.3)
with respect to z logarithmically and using (1.11), we have

1 +
(∆p,q,s[α1 + 1]f(z))

′′

(∆p,q,s[α1 + 1]f(z))
′ + p = (

α1 + 1

A1
)(

(∆p,q,s[α1 + 2]f(z))
′

(∆p,q,s[α1 + 1]f(z))
′ − 1)

(2.4) =
(B −A)(p−λ) w(z)

1 + Bw(z)
+

(B −A)(p−λ)z w
′
(z)

(1 + Bw(z)){α1 + [Bα1 + A1(B −A)(p−λ)] w(z)}
.

If we suppose now that

(2.5) max
|z|≤|z0|

|w(z)| = |w(z0)| = 1 (z0 ∈ U) ,

and apply Jack’s lemma, we find that

(2.6) z0w
′
(z0) = γw(z0) (γ ≥ 1) .

Writing

w(z0) = e
iθ (0 ≤ θ ≤ 2π)

and putting z = z0 in (2.4), we get after some computations that∣∣∣∣∣∣∣∣∣
1 +

z(∆p,q,s[α1 + 1]f(z))
′′

(∆p,q,s[α1 + 1]f(z))
′ + p

B(1 +
z(∆p,q,s[α1 + 1]f(z))

′′

(∆p,q,s[α1 + 1]f(z))
′ + p) + (A−B)(p−λ)

∣∣∣∣∣∣∣∣∣
2

− 1

=

∣∣∣∣(α1 + γA1) + [α1B + A1(B −A)(p−λ)]eiθα1 + [B(α1 −γA1) + A1(B −A)(p−λ)]eiθ
∣∣∣∣2 − 1

(2.7) =
γ2A21(1−B

2)+2γA1[α1(1+B
2)+A1B(B−A)(p−λ)]+2γA1[2α1B+A1(B−A)(p−λ)] cos θ

|α1+[B(α1−γA1)+A1(B−A)(p−λ)]eiθ|2
.



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 5

Set

g(θ) = γ2A21(1 −B
2) + 2γA1[α1(1 + B

2) + A1B(B −A)(p−λ)]

(2.8) +2γA1[2α1B + A1(B −A)(p−λ)] cos θ (0 ≤ θ ≤ 2π)

(−1 < B < A ≤ 1; 0 ≤ λ < p ; p ∈ N ; α1 ∈ R\{0}; A1 > 0; γ ≥ 1; 0 ≤ θ ≤ 2π).
Then, by hypothesis, we have

g(0) = γ2A21(1 −B
2) + 2γA1(1 + B)[α1(1 + B) + A1(B −A)(p−λ)] ≥ 0

and

g(π) = γ2A21(1 −B
2) + 2γA1(1 −B)[α1(1 −B) −A1(B −A)(p−λ)] ≥ 0

which, together, show that

(2.9) g(θ) ≥ 0 (0 ≤ θ ≤ 2π) .
In view of (2.9) , (2.7) would obviously contradict our hypothesis that f(z) ∈

∑
p(q,s,α1 + 1; A,B,λ).

Hence, we must have

(2.10) |w(z)| < 1 (z ∈ U),
and we conclude from (2.3) that

f(z) ∈
∑

p
(q,s,α1; A,B,λ).

The proof of Theorem 1 is thus completed. �

Next we prove an inclusion property associated with a certain integral transform introduced below.

Theorem 2. . Let µ be a complex number such that

<(µ) >
(A−B)(p−λ)

1 + B
(−1 < B < A ≤ 1 ; 0 ≤ λ < p ; p ∈ N).

If f(z) ∈
∑
p(q,s,α1; A,B,λ) , then the function F(z) defined by

(2.11) F(z) =
µ

zµ+p

z∫
0

tµ+p−1f(t) dt

also belongs to the class
∑
p(q,s,α1; A,B,λ) .

Proof. From (2.11), we readily have

(2.12) z(∆p,q,s[α1])F(z))
′

= µ ∆p,q,s[α1]f(z) − (µ + p)∆p,q,s[α1]F(z).
Suppose that f(z) ∈

∑
p(q,s,α1; A,B,λ) and put

(2.13)
α1
A1

(
(∆p,q,s[(α1 + 1)]F(z))

′

(∆p,q,s[α1]F(z))
′ − 1) =

(B −A)(p−λ) w(z)
1 + Bw(z)

,

where the function w(z) is either analytic or meromorphic in U, with w(0) = 0. Then, by using
(2.12), (2.13) and the identity (1.11), we find after some calculations that



6 MURUGUSUNDARAMOORTHY AND AOUF

α1
A1

(
(∆p,q,s[α1 + 1]f(z))

′

(∆p,q,s[α1]f(z))
′ − 1)

(2.14) =
(B −A)(p−λ)w(z)

1 + Bw(z)
+

(B −A)(p−λ)z w
′
(z)

(1 + Bw(z)){µ + [µB + (B −A)(p−λ)]w(z)}
.

The remaining part of the proof of Theorem 2 is similar to that Theorem 1.We choose to omit the details
involved . �

3. . Properties of the class
∑+
p (q,s,α1; A,B,λ)

In this section we assume further that :

αn, An > 0(n = 1, ...,q), βn, Bn > 0 (n = 1, ...,s), 0 ≤ B < 1, 0 ≤ λ < pand p ∈ N.
We first determine a necessary and sufficient condition for a function f(z) ∈ Σp of the form (1.15) to be
in the class

∑+
p (q,s,α1; A,B,λ) of meromorphically p-valent functions with positive coefficients.

Theorem 3. . Let f(z) ∈ Σp be given by (1.15). Then f(z) ∈
∑+
p (q,s,α1; A,B,λ) if and only if

(3.1)

∞∑
k=p

Ωkσk+p(α1)[(k + p)(1 + B) + (A−B)(p−λ)]|ak| ≤ p(A−B)(p−λ) ,

where, for convenience,

(3.2) σm(α1) =
Γ(α1 + mA1).....Γ(αq + mAq)

Γ(β1 + mB1).....Γ(βs + mBs)m!
.

Proof. Let f(z) ∈
∑+
p (q,s,α1; A,B,λ) is given by (1.15 ). Then from (1.14 ) and (1.15 ), we have∣∣∣∣∣ z(∆p,q,s[α1]f(z))

′′
+ (1 + p)(∆p,q,s[α1]f(z))

′

B (z(∆p,q,s[α1]f(z))
′′

+ (1 + p)(∆p,q,s[α1]f(z))
′
) + (A−B)(p−λ)(∆p,q,s[α1]f(z))

′

∣∣∣∣∣
=

∣∣∣∣∣∣∣∣
∞∑
k=p

Ωk(k + p)σk+p(α1)|ak|zk+p

p(A−B)(p−λ) −
∞∑
k=p

[B(k + p) + (A−B)(p−λ)]ΩkΓk+p(α1)|ak|zk+p

∣∣∣∣∣∣∣∣
< 1 (z ∈ U).

Since |<(z)| ≤ |z|(z ∈ C), we have

<




∞∑
k=p

Ωk(k + p)σk+p(α1)|ak|zk+p

p(A−B)(p−λ) −
∞∑
k=p

[B(k + p) + (A−B)(p−λ)]Ωkσk+p(α1)|ak|zk+p




(3.3) < 1 (z ∈ U).
We consider real values of z and take z = r with 0 ≤ r < 1 . Then , for
r = 0, the denominator of (3.3) is positive and so is positive for all r(0 < r < 1). Letting z = r →
1−, (3.3) yields

∞∑
k=p

Ωk(k + p)σk+p(α1)|ak| ≤ p(A−B)(p−λ)



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 7

−
∞∑
k=p

[B(k + p) + (A−B)(p−λ)]Ωkσk+p(α1)|ak|,

which leads us at once to (3.1).
In order to prove the converse, we assume that the inequality (3.1) holds true. Then we get∣∣∣∣∣∣∣∣∣

1 +
z(∆p,q,s[α1]f(z))

′′

(∆p,q,s[α1]f(z))
′ + p

B(1 +
z(∆p,q,s[α1]f(z))

′′

(∆p,q,s[α1]f(z))
′ + p) + (A−B)(p−λ)

∣∣∣∣∣∣∣∣∣
≤

∞∑
k=p

Ωk(k + p)σk+p(α1)|ak|

p(A−B)(p−λ) −
∞∑
k=p

[B(k + p) + (A−B)(p−λ)]Ωkσk+p(α1)|ak|

< 1 ( z ∈ U).

Hence, by the maximum modulus theorem, we have f(z) ∈
∑+
p (q,s,α1; A,B,λ). This completes the

proof of Theorem 3.

Corollary 1. . Let f(z) ∈ Σp be given by ( 1.15 ). If f(z) ∈
∑+
p (q,s,α1; A,B,λ), then

(3.4) |ak| ≤
p(A−B)(p−λ)

Ωkσk+p(α1)[(k + p)(1 + B) + (A−B)(p−λ)]
(k ≥ p; p ∈ N) .

The result is sharp for the function f(z) given by

(3.5) f(z) = z−p +
p(A−B)(p−λ)

Ωkσk+p(α1)[(k + p)(1 + B) + (A−B)(p−λ)]
zk (k ≥ p; p ∈ N).

Putting λ = 0 , An = 1(n = 1, ...,q) and Bn = 1(n = 1, ...,s) in Theorem 3 , we obtain

Corollary 2. .Let f(z) ∈ Σp be given by (1.15). Then f(z) ∈ Ω+p,q,s(α1; A,B), if and only if

(3.6)

∞∑
k=p

kΓk+p(α1)[(k + p)(1 + B) + p(A−B)] |ak| ≤ p2(A−B),

where

(3.7) Γm(α1) =
(α1)m.......(αq)m

(β1)m.......(βs)mm!
(m ∈ N),

and (θ)γ is the Pochhammer symbol defined, in terms of the Gamma function Γ by

(θ)γ =
Γ(θ + ν)

Γ(θ)
=

{
1 , (ν = 0 ; θ ∈ C/{0})
θ(θ + 1)....(θ + ν − 1) , (ν ∈ N ; θ ∈ C).

Remark 1. .We note the result obtained by Liu and Srivastava [20 , Theorem 3 ] is not correct . The
correct result is given by Corollary 2 .

Next we prove the following growth and distortion properties for the class
∑+
p (q,s,α1; A,B,λ).

Theorem 4. . Let the function f(z) of the form (1.15) belong to the class
∑+
p (q,s,α1; A,B,λ) . If the

sequence {δk} is nondecreasing, then



8 MURUGUSUNDARAMOORTHY AND AOUF

(3.8) r−p −
p(A−B)(p−λ)

Ωδp
rp ≤ |f(z)| ≤ r−p +

p(A−B)(p−λ)
Ωδp

rp (0 < |z| = r < 1),

where

(3.9) δk = kσk+p (α1) [(k + p)(1 + B) + (A−B)(p−λ)] (k ≥ p ; p ∈ N)

and σk+p(α1) is given by (3.2). If the sequence {δkk } is nondecreasing, then

pr−p−1 −
p2(A−B)(p−λ)

Ωδp
rp−1 ≤

∣∣∣f′ (z)∣∣∣ ≤ pr−p−1 + p2(A−B)(p−λ)
Ωδp

rp−1

(3.10) (0 < |z| = r < 1).

Each of these results is sharp with the extremal function f(z) given by

(3.11) f(z) = z−p +
(A−B)(p−λ)

Ωσ2p (α1)[2p(1 + B) + (A−B)(p−λ)]
zp (p ∈ N).

Proof. Let the function f(z) , given by (1.15), be in the class
∑+
p (q,s,α1; A,B,λ). If the sequence {δk} is

nondecreasing and positive, then , by Theorem 3 , we have

(3.12)

∞∑
k=p

|ak| ≤
p(A−B)(p−λ)

Ωδp

and if the sequence {δk
k
} is nondecreasing and positive , Theorem 3 also yields

(3.13)

∞∑
k=p

k |ak| ≤
p2(A−B)(p−λ)

Ωδp
.

Making use of the conditions (3.12) and (3.13), in conjunction with the definition (1.15), we readily obtain
the assertions (3.8) and (3.10) of Theorem 4.

Finally , it is easy to see that the bounds in (3.8) and (3.10) are attained for the function f(z) given
by (3.11).

Next we determine the radii of meromorphically p-valent starlikeness of order ϕ(0 ≤ ϕ < p) for
functions in the class

∑+
p (q,s,α1; A,B,λ).

Theorem 5. .Let the function f(z) defined by (1.15) be in the class
∑+
p (q,s,α1; A,B,λ). Then , f(z)

is meromorphically p-valent starlike of order ϕ(0 ≤ ϕ < p) in the disc |z| < r1 , that is ,

(3.14) <{−
zf

′
(z)

f(z)
} > ϕ (|z| < r1; 0 ≤ ϕ < p ; p ∈ N),

where

(3.15) r1 = inf
k≥p

{
Ωkσk+p(α1)(p−ϕ)[(k + p)(1 + B) + (A−B)(p−λ)])

p(A−B)(p−λ)(k + ϕ)

} 1
k + p

.

and σk+p(α1) is given by (3.2). The result is sharp for the function f(z) given by (3.5).



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 9

Proof. From the definition (1.15), we easily get

(3.16)

∣∣∣∣∣∣∣∣∣
zf

′
(z)

f(z)
+ p

zf
′
(z)

f(z)
−p + 2ϕ

∣∣∣∣∣∣∣∣∣
≤

∞∑
k=p

(k + p)|ak||z|k+p

2(p−ϕ) −
∞∑
k=p

(k −p + 2ϕ)|ak||z|k+p
.

Thus, we have the desired inequality :

(3.17)

∣∣∣∣∣∣∣∣∣
zf

′
(z)

f(z)
+ p

zf
′
(z)

f(z)
−p + 2φ

∣∣∣∣∣∣∣∣∣
≤ 1 (0 ≤ ϕ < p; p ∈ N)

if

(3.18)

∞∑
k=p

(
k + ϕ

p−ϕ
) |ak||z|k+p ≤ 1 .

Hence, by Theorem 3, (3.18) will be true if

(3.19) (
k + ϕ

p−ϕ
) |z|k+p ≤

Ωkσk+p(α1)[(k + p)(1 + B) + (A−B)(p−λ)]
p(A−B)(p−λ)

(k ≥ p; p ∈ N) .

The last inequality (3.19) leads us immediately to the disc |z| < r1, where r1 is given by (3.15).

4. . Neighborhoods

In this section , we also assume that

αn,An > 0 (n = 1, ...,q) and βn,Bn > 0 (n = 1, ...,s)

and

1 +

s∑
n=1

Bn −
s∑

n=1

An ≥ 0.

Following the earlier works (based upon the familiar concept of neighorhoods of analytic functions) by
Goodman [12] and Ruscheweyh [30], and (more recenlty) by Altintas et al. ([1],[2] and [3]), Liu [17], and
Liu and Srivastava ([19], [20] and [21]), we begin by introducing here the δ-neighborhood of a function
f(z) ∈ Σp of the form (1.1) by means of the definition given below :

Nδ(f) =

{
g : g ∈ Σp , g(z) = z−p +

∞∑
k=1

bkz
k−p and

∞∑
k=1

Ω(k + p)σk(α1)[(A−B)(p−λ) + k(1 + |B|)]
p(A−B)(p−λ)

|ak − bk| ≤ δ

(4.1) (−1 ≤ B < A ≤ 1; δ > 0 ; 0 ≤ λ < p; p ∈ N)} .

Making use of the definition (4.1), we now prove Theorem 6 below.

Theorem 6. . Let the function f(z) defined by (1.1) be in the class
∑
p(q,s,α1; A,B,λ). If f(z) satisfies

the following condition :



10 MURUGUSUNDARAMOORTHY AND AOUF

(4.2)
f(z) + �z−p

1 + �
∈
∑

p
(q,s,α1; A,B,λ) (� ∈ C, |�| < δ,δ > 0) ,

then

(4.3) Nδ(f) ⊂
∑

p
(q,s,α1; A,B,λ) .

Proof. It is easily seen from (1.14) that g(z) ∈
∑
p(q,s,α1; A,B,λ) if and only if, for any complex ζ with

|ζ| = 1,

(4.4)

1 +
z(∆p,q,s[α1]g(z))

′′

(∆p,q,s[α1]g(z))
′ + p

B(1 +
z(∆p,q,s[α1]g(z))

′′

(∆p,q,s[α1]g(z))
′ ) + [pB + (A−B)(p−λ)]

6= ζ

(z ∈ U; ζ ∈ C; |ζ| = 1),
which is equivalent to

(4.5)
(g ∗h)(z)
z−p

6= 0 (z ∈ U) ,

where, for convenience,

h(z) = z−p +

∞∑
k=1

ckz
k−p

(4.6) = z−p +

∞∑
k=1

Ω(k −p)σk(α1)[(A−B)(p−λ)ζ + k(Bζ − 1)]
pζ(B −A)(p−λ)

zk−p .

From (4.6), we have

|ck| =
∣∣∣∣Ω(k −p)σk(α1)[(A−B)(p−λ)ζ + k(Bζ − 1)]pζ(B −A)(p−λ)

∣∣∣∣
(4.7) ≤

Ω(k + p)σk(α1)[(A−B)(p−λ) + k(1 + |B|)]
p(A−B)(p−λ)

(k,p ∈ N) .

Now , if

f(z) = zp +

∞∑
k=1

akz
k−p ∈ Σp

satisfies the condition (4.2), then (4.5) yields

(4.8)

∣∣∣∣(f ∗h)(z)z−p
∣∣∣∣ ≥ δ (z ∈ U; δ > 0) .

By letting

(4.9) g(z) = z−p +

∞∑
k=1

bkz
k−p ∈ Nδ(f) ,

so that

∣∣∣∣[f(z) −g(z)] ∗h(z)z−p
∣∣∣∣ =

∣∣∣∣∣
∞∑
k=1

(ak − bk)ckzk
∣∣∣∣∣



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 11

≤ |z|
∞∑
k=1

Ω(k + p)σk(α1)[(A−B)(p−λ) + k(1 + |B|)]
p(A−B)(p−λ)

|ak − bk|

(4.10) < δ (z ∈ U; δ > 0),
which leads us to (4.5), and hence also (4.4) for any ζ ∈ C such that |ζ| = 1. This implies that g(z) ∈∑
p(q,s,α1; A,B,λ), which evidenlty completes the proof of the assertion (4.3) of Theorem 6.

We now define the δ-neighorhood of a function f(z) ∈ Σp of the form (1.15) as follows

N+δ (f) =


g : g ∈ Σp , g(z) = z−p +

∞∑
k=p

|bk|zk and

∞∑
k=p

Ωkσk+p(α1)[(A−B)(p−λ) + (k + p)(1 + B)]
p(A−B)(p−λ)

||ak|− |bk|| ≤ δ

(4.11) (0 ≤ B < A ≤ 1 ; δ > 0 ; 0 ≤ λ < p; p ∈ N)} .

Theorem 7. . Let the function f(z) defined by (1.13) be in the class
∑+
p (q,s,α1 + 1; A,B,λ)(0 ≤ B <

A ≤ 1 ; 0 ≤ λ < p; p ∈ N). Then

(4.12) N+δ (f) ⊂
∑+

p
(q,s,α1; A,B,λ) (δ =

2p

α1 + 2p
) .

The result is sharp in the sense that δ cannot be increased .

Proof. Making use of the same method as in the proof of Theorem 6, we can show that [cf. Equation
(4.6)]

h(z) = z−p +
∞∑
k=p

ckz
k

(4.13) = z−p +

∞∑
k=p

Ωkσk+p(α1)[(A−B)(p−λ)ζ + (k + p)(Bζ − 1)]
pζ(B −A)(p−λ)

zk .

If f(z) ∈
∑+
p (q,s,α1 + 1; A,B,λ) is given by (1.15), we obtain

∣∣∣∣(f ∗h)(z)z−p
∣∣∣∣ =

∣∣∣∣∣∣1 +
∞∑
k=p

ck|ak|zk+p
∣∣∣∣∣∣

≥ 1 −
α1

α1 + 2p

∞∑
k=p

kσk+p(α1 + 1)[(A−B)(p−λ) + (k + p)(1 + B)]
p(A−B)(p−λ)

|ak|

≥ 1 −
α1

α1 + 2p
=

2p

α1 + 2p
= δ,

by appealing to assertion (3.1) of Theorem 3. The remaining part of our proof of Theorem 7 is similar
to that of Theorem 6, and we skip the details involed.

To show sharpness of the assertion Theorem 7 , we consider the functions f(z) and g(z) given by

(4.14) f(z) = z−p +
(A−B)(p−λ)

Ωσ2p(α1 + 1)[(A−B)(p−λ) + 2p(1 + B)]
zp ∈

∑+
p

(q,s,α1 + 1; A,B,λ)

and



12 MURUGUSUNDARAMOORTHY AND AOUF

g(z) = z−p +

[
(A−B)(p−λ)

Ωσ2p(α1 + 1)[(A−B)(p−λ) + 2p(1 + B)]
+

(4.15)
(A−B)(p−λ)δ

′

Ωσ2p(α1)[(A−B)(p−λ) + 2p(1 + B)]

]
zp,

where

δ
′
> δ =

2p

α1 + 2p
.

Clearly, the function g(z) belongs to N+
δ
′ (f). On the other hand, we find from Theorem 3 that g(z) is

not in the class
∑+
p (q,s,α1; A,B,λ). Thus the proof of Theorem 7 is completed.

Finally , we prove the following theorem.

Theorem 8. . Let f(z) ∈ Σp be given by (1.1) and define the partial sums s1(z) and sn(z) as follows :

(4.16) s1(z) = z
−p and sn(z) = z

−p +

n−1∑
k=1

akz
k−p (n ∈ N) ,

it being understood that an empty sum is (as usual) nil. Suppose also that

(4.17)

∞∑
k=1

dk|ak| ≤ 1 (dk =
Ω(k + p)σk(α1)[(A−B)(p−λ) + k(1 + |B|)]

p(A−B)(p−λ)
) .

Then
(i) f(z) ∈

∑
p(q,s,α1; A,B,λ),

(ii) If {σk(α1)}(k ∈ N) is nondecreasing and

(4.18) σ1(α1) >
p(A−B)(p−λ)

Ω(1 + p)[(A−B)(p−λ) + (1 + |B|)]
,

then

(4.19) <
{
f(z)

sn(z)

}
> 1 −

1

dn
(z ∈ U ; n ∈ N) ,

and

(4.20) <
{
sn(z)

f(z)

}
>

dn
1 + dn

(z ∈ U ; n ∈ N) .

Each of the bounds in (4.19) and (4.20) is the best possible for each n ∈ N.

Proof. (i) It is not difficult to see that z−p ∈
∑
p(q,s,α1; A,B,λ)(p ∈ N). Thus, from Theorem 6 and

the hypothesis (4.17) of Theorem 8, we have

(4.21) N1(z
−p) ⊂

∑
p
(q,s,α1; A,B,λ) (0 ≤ λ < p; p ∈ N),

which shows that f(z) ∈
∑
p(q,s,α1; A,B,λ) .

(ii) Under the hypothesis in Part (ii) of Theorem 8, we can see from (4.17) that

(4.22) dk+1 > dk > 1 (k ∈ N) .



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 13

Therefore, we have

(4.23)

n−1∑
k=1

|ak| + dn
∞∑
k=n

|ak| ≤
∞∑
k=1

dk|ak| ≤ 1 ,

where we have used the hypothesis (4.17) again.
By setting

(4.24) g1(z) = dn

[
f(z)

sn(z)
− (1 −

1

dn
)

]
= 1 +

dn
∞∑
k=n

akz
k

1 +
n−1∑
k=1

akzk
,

and applying (4.23), we find that

(4.25)

∣∣∣∣g1(z) − 1g1(z) + 1
∣∣∣∣ ≤

dn
∞∑
k=n

|ak|

2 − 2
n−1∑
k=1

|ak|−dn
∞∑
k=n

|ak|
≤ 1 (z ∈ U),

which readily yields the assertion (4.19) of Theorem 8. If we take

(4.26) f(z) = z−p −
zn−p

dn
,

then

f(z)

sn(z)
= 1 −

zn

dn
→ 1 −

1

dn
( z → 1−) ,

which shows that the bound in (4.19) is the best possible for each n ∈ N.
Similarly, if we put

(4.27) g2(z) = (1 + dn)(
sn(z)

f(z)
−

dn
1 + dn

) = 1 −
(1 + dn)

∞∑
k=n

akz
k

1 +
∞∑
k=1

akzk

and make use of (4.23), we can deduce that

(4.28)

∣∣∣∣g2(z) − 1g2(z) + 1
∣∣∣∣ ≤

(1 + dn)
∞∑
k=n

|ak|

2 − 2
n−1∑
k=1

|ak| + (1 −dn)
∞∑
k=n

|ak|
≤ 1 (z ∈ U),

which leads us immediately to the assertion (4.20) of Theorem 8.
The bound in (4.20) is sharp for each n ∈ N, with the extremal function f(z) given by (4.26). The

proof of Theorem 8 is thus completed.

5. . Convolution properties

For the functions

(5.1) fj(z) = z
−p +

∞∑
k=p

|ak,j|zk (j = 1, 2; p ∈ N) ,



14 MURUGUSUNDARAMOORTHY AND AOUF

we denote by (f1 ∗f2)(z) the Hadamard product (or convolution) of the functions f1(z) and f2(z), that
is,

(5.2) (f1 ∗f2)(z) = z−p +
∞∑
k=p

|ak,1||ak,2|zk .

Throughout this section, we assume further that the sequence {σm(α1)}(m ∈ N) is nondecreasing , where
σm(α1) is given by (3.2),

(5.3) C(p,λ,A,B,k) = (k + p)(1 + B) + (A−B)(p−λ) (k ≥ p)
and

(5.4) D(p,λ,A,B) = p(A−B)(p−λ) .

Theorem 9. . Let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class
∑+
p (q,s,α1; A,B,λ).

Then (f1 ∗f2)(z) ∈
∑+
p (q,s,α1; A,B,γ), where

(5.5) γ = p(1 −
2(1 + B)(A−B)(p−λ)2

Ωσ2p(α1)[2p(1 + B) + (A−B)(p−λ)]2 − (A−B)2(p−λ)2
).

The result is sharp for the functions fj(z)(j = 1, 2) given by

(5.6) fj(z) = z
−p +

(A−B)(p−λ)
Ωσ2p(α1)[2p(1 + B) + (A−B)(p−λ)]

zp (j = 1, 2; p ∈ N) .

Proof. Employing the technique used earlier by Schild and Silverman [31], we need to find the largest γ
such that

(5.7)

∞∑
k=p

Ωkσk+p(α1)C(p,γ,A,B,k)

D(p,γ,A,B)
|ak,1||ak,2| ≤ 1

for fj(z) ∈
∑+
p (q,s,α1; A,B,λ)(j = 1, 2). Since fj(z) ∈

∑+
p (q,s,α1; A,B,λ)(j = 1, 2), we readily see

that

(5.8)
∞∑
k=p

Ωkσk+p(α1)C(p,λ,A,B,k)

D(p,λ,A,B)
|ak,j| ≤ 1 (j = 1, 2).

Therefore, by the Cauchy-Schwarz inequality, we obtain

(5.9)

∞∑
k=p

Ωkσk+p(α1)C(p,λ,A,B,k)

D(p,λ,A,B)

√
|ak,1||ak,2| ≤ 1 .

This implies that we only need to show that

(5.10)
C(p,γ,A,B,k)

(p−γ)
|ak,1||ak,2| ≤

C(p,λ,A,B,k)

(p−λ)

√
|ak,1||ak,2| (k ≥ p)

or, equivalently, that

(5.11)
√
|ak,1||ak,2| ≤

(p−γ)C(p,λ,A,B,k)
(p−λ)C(p,γ,A,B,k)

(k ≥ p).



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 15

Hence, by the inequality (5.9), it is sufficient to prove that

(5.12)
D(p,λ,A,B)

Ωkσk+p(α1)C(p,λ,A,B,k)
≤

(p−γ)C(p,λ,A,B,k)
(p−λ)C(p,γ,A,B,k)

(k ≥ p) .

It follows from (5.12) that

(5.13) γ ≤ p−
p(k + p)(1 + B)(A−B)(p−λ)2

Ωkσp+k(α1)[C(p,λ,A,B,k)]2 −p(A−B)2(p−λ)2
(k ≥ p).

Now, defining the function Φ(k) by

(5.14) Φ(k) = p−
p(k + p)(1 + B)(A−B)(p−λ)2

Ωkσp+k(α1)[C(p,λ,A,B,k)]2 −p(A−B)2(p−λ)2
(k ≥ p) ,

we see that Φ(k) is an increasing function of k. Therefore, we conclude that

(5.15) γ ≤ Φ(p) = p(1 −
2(1 + B)(A−B)(p−λ)2

Ωσ2p(α1)[2p(1 + B) + (A−B)(p−λ)]2 − (A−B)2(p−λ)2
),

which evidently completes the proof of Theorem 9.

Putting A = β and B = −β (0 < β ≤ 1) in Theorem 9, we obtain the following consequence.

Corollary 3. . Let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class
∑+
p (q,s,α1; λ,β). Then

(f1 ∗f2)(z) ∈
∑+
p (q,s,α1; γ,β), where

(5.16) γ = p(1 −
β(1 −β)(p−λ)2

Ωσ2p(α1)(p−λβ)2 −β2(p−λ)2
) .

The result is sharp for the functions fj(z)(j = 1, 2) given by

(5.17) fj(z) = z
−p +

β(p−λ)
Ωσ2p(α1)(p−λβ)

zp (j = 1, 2; p ∈ N) .

Using arguments similar to those in the proof of Theorem 9, we obtain the following result .

Theorem 10. . Let the function f1(z) defined by (5.1) be in the class
∑+
p (q,s,α1; A,B,λ). Suppose

also that the function f2(z) defined by (5.1) be in the class
∑+
p (q,s,α1; A,B,γ). Then (f1 ∗ f2)(z) ∈∑+

p (q,s,α1; A,B,ξ), where

ξ = p(1 −
2(1 + B)(A−B)(p−λ)(p−γ)

Ωσ2p(α1)[2p(1 + B) + (A−B)(p−λ)][2p(1 + B) + (A−B)(p−γ)] −M
)

(5.18) (M = (A−B)2(p−λ)(p−γ)) .
The result is sharp for the functions fj(z)(j = 1, 2) given by

(5.19) f1(z) = z
−p +

(A−B)(p−λ)
Ωσ2p(α1)[2p(1 + B) + (A−B)(p−λ)]

zp (p ∈ N)

and

(5.20) f2(z) = z
−p +

(A−B)(p−γ)
Ωσ2p(α1)[2p(1 + B) + (A−B)(p−γ)]

zp (p ∈ N) .



16 MURUGUSUNDARAMOORTHY AND AOUF

Putting A = β and B = −β (0 < β ≤ 1) in Theorem 10, we obtain Corollary 4 below.

Corollary 4. . Let the function f1(z) defined by (5.1) be in the class
∑+
p (q,s,α1; λ,β). Suppose also that

the function f2(z) defined by (5.1) be in the class
∑+
p (q,s,α1; γ,β). Then (f1∗f2)(z) ∈

∑+
p (q,s,α1; η,β),

where

(5.21) η = p(1 −
β(1 −β)(p−λ)(p−γ)

Ωσ2p(α1)(p−λβ)(p−γβ) −β2(p−λ)(p−γ)
) .

The result is the best possible for the functions fj(z)(j = 1, 2) given by

(5.22) f1(z) = z
−p +

β(p−λ)
Ωσ2p(α1)(p−λβ)

zp (p ∈ N)

and

(5.23) f2(z) = z
−p +

β(p−γ)
Ωσ2p(α1)(p−γβ)

zp (p ∈ N).

Theorem 11. . Let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class
∑+
p (q,s,α1; A,B,λ).

Then the function h(z) defined by

(5.24) h(z) = z−p +

∞∑
k=p

(|ak,1|2 + |ak,2|2)zk

belongs to the class
∑+
p (q,s,α1; A,B,ζ), where

(5.25) ζ = p(1 −
4(1 + B)(A−B)(p−λ)2

Ωσ2p(α1)[2p(1 + B) + (A−B)(p−λ)]2 − 2(A−B)2(p−λ)2
) .

This result is sharp for the functions fj(z)(j = 1, 2) given already by (5.6).

Proof. Noting that

∞∑
k=p

[Ωkσk+p(α1)C(p,λ,A,B,k)]
2

[D(p,λ,A,B)]2
|ak,j|2

(5.26) ≤ (
∞∑
k=p

Ωkσk+p(α1)C(p,λ,A,B,k)

D(p,λ,A,B)
|ak,j|)2 ≤ 1 (j = 1, 2) ,

for fj(z) ∈
∑+
p (q,s,α1; A,B,λ)(j = 1, 2), we have

(5.27)

∞∑
k=p

[Ωkσk+p(α1)C(p,λ,A,B,k)]
2

2[D(p,λ,A,B)]2
(|ak,1|2 + |ak,2|2) ≤ 1 .

Therefore, we have to find the largest ζ such that

(5.28)
C(p,ζ,A,B,k)

(p− ζ)
≤

Ωkσk+p(α1)[C(p,λ,A,B,k)]
2

2p(A−B)(p−λ)2
(k ≥ p) ,

that is, that



FAMILIES OF MEROMORPHIC MULTIVALENT FUNCTIONS 17

(5.29) ζ ≤ p−
2p(k + p)(1 + B)(A−B)(p−λ)2

Ωkσk+p(α1)[C(p,λ,A,B,k)]2 − 2p(A−B)2(p−λ)2
(k ≥ p) .

Now, defining a function Ψ(k) by

(5.30) Ψ(k) = p−
2p(k + p)(1 + B)(A−B)(p−λ)2

Ωkσk+p(α1)[C(p,λ,A,B,k)]2 − 2p(A−B)2(p−λ)2
(k ≥ p) ,

we observe that Ψ(k) is an increasing function of k. We thus conclude that

(5.31) ζ ≤ Ψ(p) = p(1 −
4(1 + B)(A−B)(p−λ)2

Ωσ2p(α1)[2p(1 + B) + (A−B)(p−λ)]2 − 2(A−B)2(p−λ)2
) ,

which completes the proof of Theorem 11.

Putting A = β and B = −β (0 < β ≤ 1) in Theorem 11, we obtain the following corollary.

Corollary 5. . Let the functions fj(z)(j = 1, 2) defined by (5.1) be in the class
∑+
p (q,s,α1 ; λ,β). Then

the function h(z) defined by (5.24) belongs to the class
∑+
p (q,s,α1 ; τ,β), where

(5.32) τ = p(1 −
2β(1 −β)(p−λ)2

Ωσ2p(α1)(p−λβ)2 − 2β2(p−λ)2
) .

The result is sharp for the functions fj(z)(j = 1, 2) given already by (5.17).

Remark 2. .We note the results obtained by Liu and Srivastava [ 21 , Theorems 4,5 and 7 ] are not
correct . The correct results are given by Theorems 4,5 and 7 , respectively , after putting λ = 0 ,An =
1(n = 1, ...,q) and Bn = 1(n = 1, ...,s).

References

[1] O. Altintas and S. Owa, Neighborhoods of certain analytic functions with negative coefficients, Internat. J. Math.

Math. Sci. 19(1996), 797-800.

[2] O. Altintas, O. Ozkan and H. M. Srivastava, Neighborhoods of a class of analytic functions with negative coefficients,
Appl. Math. Lett. 13(2000), no. 3, 63-67.

[3] O. Altintas, O. Ozkan and H. M. Srivastava, Neighborhoods of a certain family of multivalent functions with negative
coefficients, Comput. Math. Appl. 47(2004), 1667-1672.

[4] M. K. Aouf, On a class of meromorphic multivalent functions with positive coefficients, Math. Japon. 35(1990), 603-608.

[5] M. K. Aouf, A generalization of meromorphic multivalent functions with positive coefficients, Math. Japon. 35(1990),
609-614.

[6] M. K. Aouf and H. M. Hossen, New criteria for meromorphic p-valent starlike functions, Tsukuba J. Math. 17(1993),

481-486.
[7] M. K. Aouf, H. M. Hossen and H. E. Elattar, A certain class of meromorphic multivalent functions with positive and

fixed second coefficients, Punjab Univ. J. Math. 33(2000), 115-124.

[8] J. Dziok and R. K. Raina, Families of analytic functions associated with the Wright generalized hypergemetrric function
, Demonstratio Math. 37 (2004), no.3, 533-542.

[9] J. Dziok and H. M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function,

Appl. Math. Comput. 103(1999), 1-13.
[10] J. Dziok and H. M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric

function, Integral Transform. Spec. Funct. 14(2003), 7-18.

[11] A. Gangadharan, T. N. Shanmugam and H. M. Srivastava, Generalized hypergeometric functions associated with
k-uniformly convex functions, Comput. Math. Appl. 44(2002), no. 12, 1515-1526.

[12] A. W. Goodman, Univalent functions and nonanalytic curves, Proc. Amer. Math. Soc. 8(1957), 598-601.
[13] I. S. Jack, Functions starlike and convex functions of order α, J. London Math. Soc. (Ser. 2) 2(1971), no. 3, 469-474.
[14] S. B. Joshi and M. K. Aouf, Meromorphic multivalent functions with positive and fixed second coefficients, Kyungpook

Math. J. 35(1995), 163-169.
[15] S. B. Joshi and H. M. Srivastava, A certain family of meromorphically multivalent functions, Comput. Math. Appl.

38(1999), no. 3-4, 201-211.



18 MURUGUSUNDARAMOORTHY AND AOUF

[16] S. R. Kulkarni, U. H. Naik and H. M. Srivastava, A certain class of meromorphically p-valent quasi-convex functions,

PanAmer. Math. J. 8(1998), no. 1, 57-64.

[17] J. -L. Liu, Properties of some families of meromorphic p-valent functions, Math. Japon. 52(2000), no. 3, 425-434.
[18] J. -L. Liu, Strongly starlike functions associated with the Dziok-Srivastava operator, Tamkang J. Math. 35(2004), no.

1, 37-42.

[19] J. -L. Liu and H. M. Srivastava, A linear operator and associated families of meromorphically multivalent functions,
J. Math. Anal. Appl. 259(2001), 566-581.

[20] J. -L. Liu and H. M. Srivastava, Subclasses of meromorphically multivalent functions associated with a certain linear

operator, Math. Comput. Modelling 39(2004), 35-44.
[21] J. -L. Liu and H. M. Srivastava, Classes of meromorphically multivalent functions associated with the generalized

hypergeometric function, Math. Comput. Modelling 39(2004), 21-34.

[22] M. L. Mogra, Meromorphic multivalent functions with positive coefficients. I, Math. Japon. 35(1990), no. 1, 1-11.
[23] M. L. Morga, Meromorphic multivalent functions with positive coefficients. II, Math. Japon. 35(1990), no. 6, 1089-1098.

[24] S. Owa, H. E. Darwish and M. K. Aouf, Meromorphic multivalent functions with positive and fixed second coefficients,
Math. Japon. 46(1997), no. 2, 231-236.

[25] R. K. Raina, On certain classes of analytic functions and applications to fractional calculus operators , Integral

Tranform. Spec. Funct. 5(1997), 247-260..
[26] R. K. Raina and T. S. Nahar, A note on boundedness properties of Wright’s generalized hypergeometric function, Ann.

Math. Blaise Pascal 4 (1997), 83-95.

[27] R. K. Raina and T. S. Nahar, On characterization of certain Wright’s generalized hypergeometric functions involving
certain subclasses of analytic functions, Informatica 10 (1999), 219-230.

[28] R. K. Raina and T. S. Nahar, On univalent and starlike Wright’s hypergeometric functions, Rend. Sem. Math. Univ.

Padava 95 (1996) , 11-22.
[29] R. K. Raina and H. M. Srivastava, A new class of meromorphically multivalent functions with applications to generalized

hypergeometric functions, Math. Comput. Modelling 43(2006), 350-356.

[30] St. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81(1981), 521-527.
[31] A. Schild and H. Silverman, Convolution of univalent functions with negative coefficients, Ann. Univ. Mariae Curie-

Sklodowska Sect. A, 29(1975), 99-107.
[32] H. M. Srivastava H. M. Hossen and M. K. Aouf, A unified presentation of some classes of meromorphically multivalent

functions, Comput. Math. Appl. 38(1999), 63-70.

[33] H. M. Srivastava and P. W. Karlsson, Multiple Gaussian Hypergeometric Series, Halsted Press ( Ellis Horwood Ltd.,
Chichester ), John Wiley and Sons, New York , Chichester, Brisbane and London 1985.

[34] B. A. Uralegaddi and C. Somanatha, Certain classes of meromorphic multivalent functions, Tamkang J. Math. 23(1992),

223-231.
[35] B. A. Uralegaddi and C. Somanatha, Certain classes of meromorphic multivalent functions , Tamkang J. Math. 23

(1992), 223-231.

[36] E. M. Wright, The asymptotic expansion of the generalized hypergeometric function, Proc. London Math. Soc. 46
(1946), 389-408.

[37] D. -G. Yang, On new subclasses of meromorphic p-valent functions, J. Math. Res. Exposition 15(1995), 7-13.

[38] D. -G. Yang, Subclasses of meromorphic p-valent convex functions, J. Math. Res. Exposition 20(2000), 215-219.

1School of Advanced Sciences, VIT University, Vellore-632014, India

2Department of Mathematics, Mansoura University, Mansoura 35516, Egypt

∗Corresponding author