Paper-20-Acta24.2010. 109KB Jun 04 2011 12:03:12 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 24/2010
pp. 231-238

A NEW INTEGRAL OPERATOR FOR MEROMORPHIC
FUNCTIONS

Aabed Mohammed and Maslina Darus
Abstract. In this paper, a new integral operator
Rz
H(f1 , f2 , ...fn )(z) = z12 (uf1 (u))γ1 ...,(ufn (u))γn du , z ∈ U ∗
0

for fi (z) ∈ Σ is defined. In addition, starlike condition will be derived.
 ′′ Moreover,

(z)
a new subclass of meromorphic function satisfying the condition −ℜ zff (z)
+1 <

β, z ∈ U ∗ , β > 1, is introduced and sufficient conditions for this new subclass are
studied.
Keywords: Analytic and meromorphic functions, integral operators.
2000 Mathematics Subject Classification: 30C45

Introduction
Let Σ denotes the class of functions of the form
∞
P
f (z) = z1 +
ak z k ,
k=0

which are analytic and univalent in the punctured open unit disk
U ∗ = {z ∈: 0 < |z| < 1} = U − {0},
where U is the open unit disk U = {z ∈: |z| < 1}.
A function f ∈ Σ is said to be meromorphic univalent starlike of order α if

−ℜ

zf ′ (z)
>α
f (z)

(z ∈ U ∗ ; 0 ≤ α < 1),

and we denote this class by Σ∗ (α).
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A. Mohammed, M. Darus - A New Integral Operator for meromorphic functions

A function f ∈ Σ is said to be meromorphic univalent convex of order α if

zf ′′ (z)
−ℜ 1 + ′
f (z)




>α

(z ∈ U ∗ ; 0 ≤ α < 1),

and we denote this class by Σk (α).
Let A denote the class of functions f normalized by f (z) = z +

∞
P

ak z k , which are

k=2

analytic in the open unit disc U .

Let N(β) be the subclass of A, consisting of functions f (z) , which satisfy the
inequality:

 ′′

(z)
+ 1 < β, z ∈ U, β > 1.
ℜ zff (z)
This class studied by Uralegaddi etal in [1], and Owa and Srivastava in[2].
Analogous to N(β) we introduce the class ΣN (β) as the following:
A function f ∈ Σ is in the class ΣN (β) if it satisfies the inequality
 ′′

(z)
−ℜ zff (z)
+ 1 < β, z ∈ U ∗ , β > 1.
The study of integral operators has been rapidly investigated by many authors in
the field of univalent functions. Recently, various integral operators have been introduced for certain class of analytic univalent functions in the unit disk. In this study,
we follow the similar approach by introducing an integral operator for the class of
meromorphic functions Σ and study its sufficient conditions.
Definition 1. Let n ∈ N, i ∈ {1, 2, 3, ..., n} , γi > 0. We define the integral operator
H(f1 , f2 , ...fn )(z) : Σn → Σ by
1
H(f1 , f2 , ...fn )(z) = 2
z

Zz

(uf1 (u))γ1 ...,(ufn (u))γn du , z ∈ U ∗ ,

(1)

0

where fi (z) ∈ Σ.
It is easy to see that this integral operator is analytic operator.
Remark 1. For n = 1,
operator

γ1 = γ, γ2 = γ3 = ... = γn = 0, we obtain the integral

232

A. Mohammed, M. Darus - A New Integral Operator for meromorphic functions

1
G[f ](z) = 2
z

Zz

(uf (u))γ du.

(2)

0

Remark 2. For n = 1,
operator

γ1 = 1, γ2 = γ3 = ... = γn = 0, we obtain the integral
1
F (z) = 2
z

Z

z

uf (u)du.

(3)

0

Main Result
In order to prove our main results (Theorem 1 and Theorem 2 below) we shall need
the following lemma due to S. S. Miller and P. T. Mocanu [3]:
Lemma 1. Let the function Ψ : C 2 → C satisfy
ℜΨ(ix, y) ≤ 0

for all real x and for all real y, y ≤ −(1 + x2 ) 2. If p(z) = 1 + p1 z + ...
is analytic in the unit disc U = {z : z ∈ C |z| < 1} and
ℜΦ(p(x), zp′ (x)) > 0, z ∈ U,
then

ℜp(z) > 0 f or z ∈ U.

First, we prove the following starlike result of the operator H[f ](z).
Theorem 1. Let fi ∈ Σ, i ∈ {1, 2, 3, ..., n}, γi > 0.
If

n

′

−ℜ

zfi (z) X
>
γi ,
fi (z)
i=1

233

A. Mohammed, M. Darus - A New Integral Operator for meromorphic functions

then H[f ](z) ∈ Σ∗ (the class of meromorphically starlike functions), where H[f ](z)
is the integral operator define as in (1).
Proof. On successive differentiation of H[f ](z), we get
z 2 H[f ]′ (z) + 2zH[f ](z) = (z f1 (z))γ1 ...(z fn (z))γn ,
n
P

i=1

z 2 H[f ]′′ (z) + 4zH[f ]′ (z) + 2H[f ](z)) =
 ′
 Q
n
(zfj (z))γj .
γi (zfi (z))γi −1 zfi (z) + fi (z)
j=1,j6=i

Then

z 2 H[f ]′′ (z)+4zH[f ]′ (z)+2H[f ](z)
z 2 H[f ]′ (z)+2zH[f ](z)

=

n
P

γi

i=1



′

fi (z)
fi (z)

+

1
z



.

By multiplying the above expression with z we obtain

 ′
n
P
zfi (z)
z 2 H[f ]′′ (z)+4zH[f ]′ (z)+2H[f ](z)
=
γi fi (z) + 1 ,
zH[f ]′ (z)+2H[f ](z)
i=1

that is equivalent to
zH′′ [f ](z)
H[f ]′ (z)

H[f ](z)
+ 2 zH[f
]′ (z) + 4

H[f ](z)
1 + 2 zH[f
]′ (z)

Thus

=

n
X

"

γi

i=1



H[f ](z)
]′′ (z)
+
1
) − 2 zH[f
−( zH[f
H[f ]′ (z)
]′ (z) − 3
H[f ](z)
1 + 2 zH[f
]′ (z)

=−

#
′
zfi (z)
+1 .
fi (z)

n
X
i=1

γi

"

#
′
zfi (z)
+1 .
fi (z)

(4)

(5)

We define the regular function p in U by
p(z) = −

zH ′ [f ](z)
,
H[f ]

z ∈ U∗

and p(0) = 1. Differentiating p(z) logarithmically, we obtain


zH[f ]′′ (z)
zp′ (z)
=− 1+
.
p(z) −
p(z)
H[f ]′ (z)
¿From (5),(6) and (7) we obtain
234

(6)

(7)

A. Mohammed, M. Darus - A New Integral Operator for meromorphic functions

#
" ′
n
X
zfi (z)
zp′ (z)
=−
+1 .
γi
p(z) − 1 −
p(z) − 2
fi (z)

(8)

i=1

¿From (4) and (8) we get
( n


X
zp′ (z)
=ℜ −
ℜ p(z) − 1 −
γi
p(z) − 2
i=1

!)
′
zfi (z)
+1
> 0.
fi (z)

We define the function Ψ by
Ψ(u, v) = u − 1 −

v
.
u−2

In order to use Lemma 1 we must verify
 that Ψ(ix, y) ≤ 0 whenever x and y are
real numbers such that y ≤ −(1 + x2 ) 2.
We have



y
ℜΨ(ix, y) = ℜ ix − 1 −
ix − 2



1 + x2
2y
≤
−1
−
< 0.
4 + x2
4 + x2

= −1 +

Then
ℜΨ(p(z), zp′ (z)) > 0, z ∈ U.
By Lemma 1, we conclude thatℜp(z) > 0, z ∈ U, and so
−ℜ

zH ′ [f ](z)
> 0.
H[f ]

(9)

Therefore H[f ](z) ∈ Σ∗ .
Corollary 1. Let f ∈ Σ. If

−ℜ

zf ′ (z)
> γ.
f (z)

Then G[f ](z) ∈ Σ∗ (the class of meromorphically starlike functions), where G[f ](z)
is the integral operator define as in (2).
Proof. From Theorem 1, by considering n = 1,
Corollary 2. Let f ∈ Σ. If
235

γ1 = γ, γ2 = γ3 = ... = γn = 0.

A. Mohammed, M. Darus - A New Integral Operator for meromorphic functions

−ℜ

zf ′ (z)
>1
f (z)

Then F [f ](z) ∈ Σ∗ (the class of meromorphically starlike functions), where F [f ](z)
is the integral operator define as in (3).
Proof. From Corollary 1, let γ = 1.
Our next result is the following:
Theorem 2. Let fi ∈ Σ, i ∈ {1, 2, 3, ..., n}.
and let
n
P

1<

γi < 2.

i=1

If

n

′

−ℜ

zfi (z) X
>
γi .
fi (z)
i=1

Then
H[f ](z) ∈ ΣN (β), β > 1,
where H[f ](z) is the integral operator define as in (1).
Proof. From (5) we know that
−



zH[f ]′′ (z)
+1
H[f ]′ (z)



H[f ](z)

−2 zH[f ]′ (z) −3

H[f ](z)

1+2 zH[f ]′ (z)

=−

n
P

i=1

γi



′

zfi (z)
fi (z)


+1 .

Let
H[f ](z)
h(z) = − zH[f
]′ (z) .

It is easy to see from (9) and Theorem 1 that ℜ h(z) = ℜ 1/p(z) > 0.
Then we have
236

A. Mohammed, M. Darus - A New Integral Operator for meromorphic functions

−(



zH[f ]′′ (z)
+1
H[f ]′ (z)



)+2h(z)−3

1−2h(z)

=−

i=1

That is equivalent to

−



 X
n
zH[f ]′′ (z)
+
1
=
γi
H[f ]′ (z)
i=1

n
P

′

zf (z)
− i
fi (z)

!

γi



′

zfi (z)
fi (z)


+1 .

n
n
X
X
[1 − 2h(z)] + 2h(z)[
γi − 1] + 3 −
γi .
i=1

i=1

(10)

Taking real parts in (10), we get




′
n
n
n
P
P
P
zfi (z)
zH[f ]′′ (z)
γi .
γi − fi (z) [1 − 2h(z)] + 2[ γi − 1]ℜh(z) + 3 −
−ℜ H[f ]′ (z) + 1 = ℜ
i=1

i=1

i=1

Since ℜ(z) < |z| , we can write



 P
n
n
n
 zf ′ (z)

P
P
zH[f ]′′ (z)
i

−ℜ H[f ]′ (z) + 1 ≤
γi − fi (z) [1 − h(z)] + 2[ γi − 1]ℜh(z) + 3 −
γi .
i=1

i=1

i=1

Let

β=

n
P

i=1

and since



n
n
 zf ′ (z)

P
P
i

γi − fi (z) [1 − 2h(z)] + 2[ γi − 1]ℜh(z) + 3 −
γi
n
P

i=1

we can conclude that
β > 2[

n
P




 zf ′ (z)
γi − fii(z) [1 − 2h(z)] > 0

γi − 1]ℜh(z) + 3 −

i=1

and so −ℜ



zH[f ]′′ (z)
H[f ]′ (z)

i=1

i=1


+ 1 < β,

n
P

i=1

γi > 3 −

n
P

γi > 1,

i=1

β > 1. Therefore H[f ](z) ∈ ΣN (β), β > 1.

Acknowledgement: The work here is fully supported by MOHE: UKM-ST-06FRGS0107-2009.
References
[1] B.A.Uralegaddi, M.D. Ganigi, S.M. Sarangi, Univalent functions positive coefficients, Tamkang J. Math, 25(1994), 225-230.
237

A. Mohammed, M. Darus - A New Integral Operator for meromorphic functions

[2] S. Owa, H.M. Srivastava, Some generalized convolution properties associated
with certain subclasses of analytic functions, JIPAM, 3(3)(2002), Article 42, 1-13.
[3] S. S. Miller and P. T. Mocanu, Differential subordination and univalent functions, Michigan Math. J, 28 (1981), 157-171.
Aabed Mohammed and Maslina Darus
School of Mathematical Sciences,
Faculty of Science and Technology,
Universiti Kebangsaan Malaysia
43600 Bangi, Selangor D. Ehsan,
Malaysia
emails: aabedukm@yahoo.com
maslina@ukm.my

238