15_815_goyal.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 3, No. 6, 2010, 1118-1123

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE ON COMPLEX ANALYSIS: THEORY AND APPLICATIONS

DEDICATED TO PROFESSOR HARI M. SRIVASTAVA,

ON THE OCCASION OF HIS 70TH
BIRTHDAY

Quasi-Hadamard Product of Certain Meromorphic P-Valent An-

alytic Functions

S. P. Goyal1,∗, Pranay Goswami2

1 Department of Mathematics, University of Rajasthan, Jaipur-302055, India
2 Department of Mathematics, Amity University Rajasthan, Jaipur-302002, India

Abstract. In this paper, we establish certain results concerning the quasi-Hadamard product for the

classes related to meromorphic p-valent analytic functions with positive coefficients.

2000 Mathematics Subject Classifications: 30C45

Key Words and Phrases: Analytic functions, Meromorphic p-valent functions, quasi-Hadamard prod-

uct

1. Introduction

Throughout this paper, let p ∈ N = {1,2,3, . . .} and the functions of the form :

ϕ(z) = apzp −
∞
∑

n=1

an+pzn+p
�

ap > 0; ap+n ≥ 0
�

,

ψ(z) = bpzp −
∞
∑

n=1

bn+pzn+p
�

ap > 0; bp+n ≥ 0
�

,

be analytic and p-valent in the unit disc ∆= {z : |z| < 1}. Also, let

f (z) =
ap−1

zp
+

∞
∑

n=1

an+p−1zn+p−1
�

ap > 0; ap+n ≥ 0
�

, (1)

∗Corresponding author.

Email addresses: somprg�gmail.
om (S. Goyal), pranaygoswami83�gmail.
om (P. Goswami)

http://www.ejpam.com 1118 c© 2010 EJPAM All rights reserved.



S. Goyal, P. Goswami / Eur. J. Pure Appl. Math, 3 (2010), 1118-1123 1119

fi(z) =
ap−1,i

zp
+

∞
∑

n=1

an+p−1,iz
n+p−1

�

ap,i > 0; ap+n,i ≥ 0
�

, (2)

g(z) =
bp−1

zp
+

∞
∑

n=1

bn+p−1zn+p−1
�

bp > 0; bp+n ≥ 0
�

, (3)

and

gi(z) =
bp−1,i

zp
+

∞
∑

n=1

bn+p−1,iz
n+p−1

�

bp,i > 0; bp+n,i ≥ 0
�

, (4)

be analytic and p-valent in the punctured disc ∆∗ = {z : 0< |z| < 1}.

Let
∑

S T ∗0(p,α) denote the class of functions f (z) defined by (1) and satisfy the condi-

tion

−Re

¨

1+
z f ′(z)

f (z)

«

> α, (z ∈∆∗) (5)

and
∑

C ∗0 (p,α) denote the class of functions f (z) defined by (1) and satisfy the condition

−Re

¨

z f ′(z)

f (z)

«

> α, (z ∈∆∗) (6)

where 0≤ α < p.

The quasi-Hadamard product of two or more functions has recently been defined and

used by Kumar ([7],[8], and [9]), Aouf et al. [3], Hossen [6], Darwish [4] and Sekine [12].

Accordingly, the quasi-Hadamard product of two functions ϕ(z) and ψ(z) is defined by

(ϕ ∗ψ)(z) = ap bpzp −
∞
∑

n=1

an+p bn+pzn+p (7)

Aouf [1] defined the Hadamard product of two meromorphic p-valent functions f (z) and g(z)

by

( f ∗ g)(z) =
ap−1 bp−1

zp
+

∞
∑

n=1

an+p−1 bn+p−1zn+p−1 (8)

Similarly, we can define the Hadamard product of more than two meromorphic p−valent

functions.

Let λ(z) be a fixed function of the form

λ(z) =
cp−1

zp
+

∞
∑

n=1

cn+p−1zn+p−1
�

cp > 0; cp+n ≥ 0
�

, (9)

Using the function defined by (9), we now define the following new classes



S. Goyal, P. Goswami / Eur. J. Pure Appl. Math, 3 (2010), 1118-1123 1120

Definition 1. A function f (z) ∈
∑

M 0
λ
(cn+p−1,δ) (cn+p−1 ≥ cp > 0; n≥ 2) if and only if

∞
∑

n=1

cn+p−1an+p−1 ≤ δap−1 (10)

where δ > 0.

Definition 2. A function f (z) ∈
∑

B k
λ
(cn+p−1,δ) (cn+p−1 ≥ cp > 0; n≥ 2) if and only if

∞
∑

n=1

�

n+ p− 1

p

�k

cn+p−1an+p−1 ≤ δap−1 (11)

where δ > 0.

It is easy to check that various subclasses of meromorphic and multivalent functions can

be (studied by various authors) represented as
∑

B k
λ
(cn+p−1,δ) for suitable choices of cn,δ

and k. For example:

(1)
∑

B k
λ
((n+ 2p− 1) + β(n+ 2α− 1), 2β(p−α))≡

∗
∑

k

(p,α,β)

(2)
∑

B0
λ
((n+ 2p− 1)+ β(n+ 2α− 1), 2β(p−α))≡

∑

S∗0(p,α,β)

(3)
∑

B1
λ
((n+ 2p− 1)+ β(n+ 2α− 1), 2β(p−α))≡

∑

C∗0(p,α,β)

(4)
∑

B k
λ
((n(1+ β) + (2α− 1)β + 1, 2β(1−α))≡

∑

S∗0(k,α,β) for p = 1

(5)
∑

B k
λ
(n(n(1+ β) + (2α− 1)β + 1), 2β(1−α))≡

∑

C∗0(k,α,β) for p= 1

The classes
∗
∑

k

(p,α,β),
∑

S∗0(p,α,β) and
∑

C∗0(p,α,β) have been studied by Aouf [1] and

the classes
∑

S∗0(k,α,β) and
∑

C∗0(k,α,β) have been studied by El-Ashwah and Aouf [5].

Evidently,
∑

B0
λ
(cn+p−1,δ) ≡

∑

M 0
λ
(cn+p−1,δ) . Further,

∑

B k
λ
(cn+p−1,δ) ⊂

∑

Bh
λ
(cn+p−1,δ)

if k > h ≥ 0, the containment being proper. Moreover, for any positive integer k we have the

following inclusion relation
∑

B k
λ(cn+p−1,δ) ⊂

∑

B k−1
λ
(cn+p−1,δ)⊂ . . . ⊂

∑

M 0
λ (cn+p−1,δ)⊂

∑

C ∗0 (p,α) ⊂
∑

S ∗0 (p,α).

We also note that for every nonnegative real number k, the class
∑

B k
λ
(cn+p−1,δ) is nonempty

as the functions of the form

f (z) =
ap−1

zp
+

∞
∑

n=1

�

p

n+ p− 1

�k δap+n−1

cp+n−1

µn+p−1zn+p−1
�

ap > 0; ap+n ≥ 0
�

,

where ap−1 > 0, µn+p−1 ≥ 0 and
∑∞

n=1µn+p−1 ≤ 1, satisfy the inequality (11).

In this paper we establish a theorem concerning the quasi-Hadamard product of func-

tions in the classes
∑

M 0
λ
(cn+p−1,δ) and

∑

B k
λ
(cn+p−1,δ). The theorem and its applications

generalize the results obtained by Aouf [1], Mogra [11] and El-Ashwah and Aouf [5].



S. Goyal, P. Goswami / Eur. J. Pure Appl. Math, 3 (2010), 1118-1123 1121

2. Main Theorem

Theorem 1. Let the functions fi(z) defined by (2) belong to the class
∑

B k
λ
(cn+p−1,δ) for every

i = 1,2, . . . , m, and let the functions g j(z) defined by (4) belong to the class
∑

M 0
λ
(cn+p−1,δ)

for every j = 1,2, . . . ,q. If cn+p−1 ≥
�

n+p−1

p

�

δ, then the quasi-Hadamard product

f1 ∗ f2 ∗ . . . ∗ fm ∗ g1 ∗ g2 ∗ . . . ∗ gq(z) belongs to the class
∑

B (k+1)m+q−1

λ
(cn+p−1,δ).

Proof. Let h(z) := f1 ∗ f2 ∗ . . . ∗ fm ∗ g1 ∗ g2 ∗ . . . ∗ gq(z), then

h(z) =

(

m
∏

i=1

ap−1,i

q
∏

j=1

bp−1, j

)

zp
+

∞
∑

n=1







m
∏

i=1

an+p−1,i

q
∏

j=1

bn+p−1, j







zn+p−1. (12)

It is sufficient to show that

∞
∑

n=1

�

n+ p− 1

p

�m(k+1)+q−1 m
∏

i=1

an+p−1,i

q
∏

j=1

bn+p−1, j ≤ δ
m
∏

i=1

ap−1,i

q
∏

j=1

bp−1, j (13)

Since fi(z) ∈
∑

B k
λ
(cn+p−1,δ), we have

∞
∑

n=1

�

n+ p− 1

p

�k

cn+p−1an+p−1,i ≤ δap−1,i (14)

for every i = 1,2, . . . , m. Therefore,

an+p−1,i ≤
�

n+ p− 1

p

�−k
�

δ

cn+p−1

�

ap−1,i (15)

which by virtue of the condition (given with the theorem) implies that

an+p−1,i ≤
�

n+ p− 1

p

�−k−1

ap−1,i (16)

for every i = 1,2, . . . , m. Further, since g j(z) ∈
∑

M 0
λ
(cn+p−1,δ), we have

∞
∑

n=1

cn+p−1 bn+p−1, j ≤ δbp−1, j (17)

for every j = 1,2, . . . ,q. Hence we obtain

bn+p−1, j ≤
�

n+ p− 1

p

�−1

bp−1, j (18)



S. Goyal, P. Goswami / Eur. J. Pure Appl. Math, 3 (2010), 1118-1123 1122

Using (16) for i = 1,2, . . . , m, and (18) for j = 1,2, . . . ,q− 1, and (17) for j = q, we get

∞
∑

n=1











�

n+ p− 1

p

�m(k+1)+q−1

cn+p−1







m
∏

i=1

an+p−1,i

q
∏

j=1

bn+p−1, j

















≤
∞
∑

n=1

�
�

n+ p− 1

p

�m(k+1)+q−1�n+ p− 1

p

�−m(k+1)�n+ p− 1

p

�−(q−1)







m
∏

i=1

ap−1,i

q−1
∏

j=1

bn+p−1, j







cn+p−1 bn+p−1,q











=







m
∏

i=1

ap−1,i

q−1
∏

j=1

bp−1, j







 

∞
∑

n=1

cn+p−1 bn+p−1,q

!

≤ δ
m
∏

i=1

ap−1,i

q
∏

j=1

bp−1, j (by (17))

Hence h(z) ∈
∑

B (k+1)m+q−1

λ
(cn+p−1,δ). This completes the proof of the Theorem 1.

Taking k = 0 in the proof of the above theorem, we obtain

Corollary 1. Let the functions fi(z) defined by (2) and the functions g j(z) defined by (4) belong

to the class
∑

M 0
λ
(cn+p−1,δ) for every i = 1,2, . . . , m, and j = 1,2, . . . ,q. If

cn+p−1 ≥
�

n+p−1

p

�

δ, then the quasi-Hadamard product f1 ∗ f2 ∗ . . . ∗ fm ∗ g1 ∗ g2 ∗ . . . ∗ gq(z)

belongs to the class
∑

Bm+q−1

λ
(cn+p−1,δ).

Now taking into account the quasi-Hadamard product functions g1(z) ∗ g2(z) ∗ . . . ∗ gq(z)

only, in the proof of the above theorem, and using (18) for j = 1,2,3 . . . ,q− 1, and (17) for

j = m, we obtain

Corollary 2. Let the functions g j(z) defined by (4) belong to the class
∑

M 0
λ
(cn+p−1,δ) for

j = 1,2, . . . ,q. If cn+p−1 ≥
�

n+p−1

p

�

δ, then Hadamard product g1 ∗ g2 ∗ . . . ∗ gq(z) belongs to

the class
∑

Bq−1

λ
(cn+p−1,δ).

Remark 1.

(i) Putting cn+p−1 = (n+ 2p− 1)+β(n+ 2α− 1) and δ = 2β(p−α) in the above theorem,

we obtain the results obtained by Aouf [1].

(ii) Putting p = 1, cn = n((n+ 1)+β(n+ 2α− 1)) and δ = 2β(1−α) in the above theorem,

we obtain the results obtained by Mogra [11].

(iii) Putting p = 1, in Corollary 2, we obtain the results obtained by El-Ashwah and Aouf [5].



REFERENCES 1123

ACKNOWLEDGEMENTS The first author (S P G) is thankful to CSIR, New Delhi, India for

awarding Emeritius Scientist under scheme No. 21(084)/10/EMR-II.

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