Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2011, Article ID 804150, 8 pages
doi:10.1155/2011/804150
Research Article
Starlikeness Properties of a New Integral Operator
for Meromorphic Functions
Aabed Mohammed and Maslina Darus
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
43600 Bangi, Selangor D. Ehsan, Malaysia
Correspondence should be addressed to Maslina Darus, maslina@ukm.my
Received 3 March 2011; Revised 15 April 2011; Accepted 3 June 2011
Academic Editor: Pablo Gonza’lez-Vera
Copyright q 2011 A. Mohammed and M. Darus. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We define here an integral operator Hγ1 ,...,γn for meromorphic functions in the punctured open unit
disk. Several starlikeness conditions for the integral operator Hγ1 ,...,γn are derived.
1. Introduction
Let Σ denotes the class of functions of the form
fz ∞
1 an zn ,
z n0
1.1
which are analytic in the punctured open unit disk
U∗ {z ∈ C : 0 < |z| < 1} U \ {0},
1.2
where U is the open unit disk U {z ∈ C : |z| < 1}.
We say that a function f ∈ Σ is meromorphic starlike of order α 0 ≤ α < 1, and
belongs to the class Σ∗ α, if it satisfies the inequality
zf z
−R
fz
> α.
1.3
2
Journal of Applied Mathematics
A function f ∈ Σ is a meromorphic convex function of order α 0 ≤ α < 1, if f satisfies
the following inequality
zf z
−R 1 > α,
f z
1.4
and we denote this class by Σk α.
Analogous to the integral operator defined by Breaz et al. 1 on the normalized analytic functions, we now define the following integral operator on the space meromorphic
functions in the class .
Definition 1.1. Let n ∈ N, γi > 0, i ∈ {1, 2, 3, . . . , n}. We define the integral operator Hγ1 ,...,γn f1 ,
f2 , . . . , fn : Σn → Σ by
Hγ1 ,...,γn
1
f1 , . . . , fn z 2
z
z 0
−u2 f1 u
γ1
γn
· · · −u2 fn u du.
1.5
For the sake of simplicity, from now on we will write Hγ1 ,...,γn z instead of Hγ1 ,...,γn f1 , . . . ,
fn z.
By Σkp β −1 ≤ β < 1, we denote the class of functions f ∈ Σ such that
zf z
zf z
f z 2 < −R f z β − 1.
1.6
In order to derive our main results, we have to recall here the following preliminary results.
Lemma 1.2 see 2. Suppose that the function Ψ : C2 → C satisfies the following condition:
R{Ψis, t} ≤ 0,
− 1 s2
s, t ∈ R; t ≤
.
2
1.7
If the function pz 1 p1 z · · · is analytic in U and
R Ψ pz, zp z > 0,
z ∈ U,
1.8
then,
R pz > 0 z ∈ U.
1.9
Proposition 1.3 see 3. If f ∈ Σ satisfying
z zf z 3f z
> α,
−R
zf z 2fz
zf z
fz 1 < 1,
0 ≤ α < 1,
1.10
Journal of Applied Mathematics
3
then,
zf z
−R
fz
1.11
> α.
2. Starlikeness of the Operator Hγ1 ,...,γn z
In this section, we investigate sufficient conditions for the integral operator Hγ1 ,...,γn z which
is defined in Definition 1.1, to be in the class Σ∗ α, 0 ≤ α < 1.
Theorem 2.1. Let fi ∈
, γi > 0 for all i ∈ {1, . . . , n}. If
−R
zfi z
fi z
>
−1
2,
nγi
2.1
then Hγ1 ,...,γn z belongs to Σ∗ 0.
Proof. On successive differentiation of Hγ1 ,...,γn z, which is defined in 1.5, we get
γ 1
γ n
2zHγ1 ,...,γn z z2 Hγ1 ,...,γn z −z2 f1 z · · · −z2 fn z ,
z2 Hγ1 ,...,γn z 4zHγ1 ,...,γn z 2Hγ1 ,...,γn z
n
γi −z2 fi z
γi −1 −z2 fi z − 2zfi z
n j1,j /
i
i1
−z2 fj z
γ j
2.2
.
Then from 2.2, we obtain
z2 Hγ1 ,...,γn z 4zHγ1 ,...,γn z 2Hγ1 ,...,γn z
z2 Hγ1 ,...,γn z 2zHγ1 ,...,γn z
n
γi
i1
fi z 2
fi z z
.
2.3
zfi z
2 .
fi z
2.4
By multiplying 2.3 with z yield,
z2 Hγ1 ,...,γn z 4zHγ1 ,...,γn z 2Hγ1 ,...,γn z
zHγ1 ,...,γn z 2Hγ1 ,...,γn z
n
γi
i1
That is equivalent to
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭
1
n
i1
γi
zfi z
2 .
fi z
2.5
4
Journal of Applied Mathematics
Or
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
n
n
zfi z
γ
γi 1.
−
−
−
2
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭ i1 i
fi z
i1
2.6
We can write the left-hand side of 2.6, as the following:
zHγ1 ,...,γn z/Hγ1 ,...,γn z 3
− zHγ1 ,...,γn z/Hγ1 ,...,γn z
zHγ1 ,...,γn z/Hγ1 ,...,γn z 2
n
n
zfi z
− 2 γi 1.
γi − fi z
i1
i1
2.7
We define the regular function p in U by
pz −
zHγ1 ,...,γn z
Hγ1 ,...,γn z
,
2.8
and p0 1. Differentiating pz logarithmically, we obtain
−pz zHγ1 ,...,γn z
zp z
1
.
pz
Hγ1 ,...,γn z
2.9
From 2.7, 2.8, and 2.9, we obtain
n
n
zfi z
zp z
pz − 2 γi 1.
γi − −pz 2 i1
fi z
i1
2.10
Let us put
Ψu, v u v
.
−u 2
2.11
From 2.1, 2.10, and 2.11, we obtain
zf1 z
zf z
R Ψ pz, zp z γ1 −R · · · −R n
− 2 γ1 · · · γn 1
f1 z
fn z
−1
−1
> γ1
2 · · · γn
2 − 2 γ1 · · · γn 1 0.
nγ1
nγn
2.12
Journal of Applied Mathematics
5
Now, we proceed to show that
− 1 s2
s, t ∈ R; t ≤
.
2
R{Ψis, t} ≤ 0,
2.13
Indeed, from 2.11, we have
R{Ψis, t} R is t
−is 2
1 s2
2t
≤
−
< 0.
4 s2
4 s2
2.14
Thus, from 2.12, 2.14, and by using Lemma 1.2, we conclude that R{pz} > 0, and so
−R
zHγ1 ,...,γn z
Hγ1 ,...,γn z
2.15
>0
that is, Hγ1 ,...,γn z is starlike of order 0.
Theorem 2.2. For i ∈ {1, . . . , n}, let γi > 0 and fi ∈ Σk αi 0 ≤ αi < 1. If 0 <
1, Hγ1 ,...,γn z be the integral operator given by 1.5 and
n
i1 γi 1
zHγ ,...,γ z
1
n
1 < 1.
Hγ1 ,...,γn z
Then Hγ1 ,...,γn z belong to Σ∗ μ, where μ 1 −
n
i1 γi 1
− αi ≤
2.16
− αi .
Proof. Following the same steps as in Theorem 2.1, we obtain
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
n
n
zfi z
−
−
1
1
−
γ
γi .
i
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭ i1
fi z
i1
2.17
Taking the real part of both terms of the last expression, we have
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
n
n
zfi z
−R
1
1
−
γ
γi .
−R
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭ i1 i
fi z
i1
2.18
6
Journal of Applied Mathematics
Since fi ∈ Σk αi , for i ∈ {1, . . . , n}, we receive
−R
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭
>
n
n
γi αi 1 − γi .
i1
2.19
i1
Therefore,
−R
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
⎩
zHγ1 ,...,γn z
2Hγ1 ,...,γn z ⎭
>1−
n
γi 1 − αi .
2.20
i1
Using 2.16, 2.20, and applying Proposition 1.3, we get Hγ1 ,...,γn z ∈ Σ∗ μ, where μ 1 − ni1 γi 1 − αi .
Letting αi α, i ∈ {1, . . . , n} in Theorem 2.2, we get the following.
Corollary 2.3. For i ∈ {1, . . . , n}, let γi > 0 and fi ∈ Σk α 0 ≤ α < 1. If
0<
n
γi ≤
i1
1
,
1−α
2.21
Hγ1 ,...,γn be the integral operator given by 1.5 and
zHγ ,...,γ z
1
n
1 < 1.
Hγ1 ,...,γn z
Then Hγ1 ,...,γn z is starlike of order 1 − 1 − α
2.22
n
i1 γi .
Theorem 2.4. For i ∈ {1, . . . , n}, let γi > 0 and fi ∈ Σkp βi −1 ≤ βi < 1. If
0<
n
γi 1 − βi ≤ 1,
2.23
i1
Hγ1 ,...,γn z be the integral operator given by 1.5 and
zHγ ,...,γ z
1
n
1 < 1.
Hγ1 ,...,γn z
Then Hγ1 ,...,γn z is starlike of order 1 −
n
i1 γi 1
− βi .
2.24
Journal of Applied Mathematics
7
Proof. Following the same steps as in Theorem 2.1, we obtain
−
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭
n
− γi
i1
zfi z
2 1
fi z
n
n
n
zfi z
−
1
1
−
β
γi −
γ
γi βi
i
i
fi z
i1
i1
i1
n
n
zfi z
βi − 1 1 − γi 1 − βi .
γi −
f
z
i
i1
i1
2.25
We calculate the real part from both terms of the above equality and obtain
−R
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭
n
γi −R
i1
zfi z
βi
fi z
2.26
n
− 1 1 − γi 1 − βi .
i1
Since fi ∈ Σkp βi for all i ∈ {1, . . . , n}, the above relation then yields
−R
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭
n
n
zf z
i
>
2 1 − γi 1 − βi .
γi fi z
i1
i1
Because
n
i1 γi |zfi z/fi z
−R
2.27
2| ≥ 0, we obtain that
⎫
⎧ ⎨ z zHγ1 ,...,γn z 3Hγ1 ,...,γn z ⎬
⎩ zHγ1 ,...,γn z 2Hγ1 ,...,γn z ⎭
>1−
n
γi 1 − βi .
2.28
i1
Using 2.24, 2.28 and applying Proposition 1.3, we get Hγ1 ,...,γn z is a starlike function of
order 1 − ni1 γi 1 − βi .
Letting βi β, i ∈ {1, . . . , n} in Theorem 2.4, we get the following.
Corollary 2.5. For i ∈ {1, . . . , n}, let γi > 0 and fi ∈ Σkp β −1 ≤ β < 1. If
0<
n
i1
γi ≤
1
,
1−β
2.29
8
Journal of Applied Mathematics
Hγ1 ,...,γn be the integral operator given by 1.5 and
zHγ ,...,γ z
1
n
1 < 1.
Hγ1 ,...,γn z
Then Hγ1 ,...,γn z is starlike of order 1 − 1 − β
2.30
n
i1 γi .
Letting n 1, γ1 γ and f1 f in Corollary 2.5, we get the following.
Corollary 2.6. Let γ > 0, and f ∈ Σkp β −1 ≤ β < 1. If
0<γ ≤
1
,
1−β
2.31
Hγ z be the integral operator,
1 z 2 γ
−u f u du,
Hγ z 2
z 0
zHγ z
1 < 1.
Hγ z
2.32
Then Hγ z is starlike of order 1 − 1 − βγ.
Other work related to integral operator for different studies can also be found in 4–6.
Acknowledgments
The work here was supported by MOHE Grant: UKM-ST-06-FRGS0244-2010. The authors
also would like to thank the referee for his/her careful reading and making some valuable
comments which have improved the presentation of this paper.
References
1 D. Breaz, S. Owa, and N. Breaz, “A new integral univalent operator,” Acta Universitatis Apulensis, no.
16, pp. 11–16, 2008.
2 S. S. Miller and P. T. Mocanu, “Differential subordinations and univalent functions,” The Michigan
Mathematical Journal, vol. 28, no. 2, pp. 157–172, 1981.
3 A. Mohammed and M. Darus, “New criteria for meromorphic functions,” Submitted.
4 A. Mohammed and M. Darus, “New properties for certain integral operators,” International Journal of
Mathematical Analysis, vol. 4, no. 41–44, pp. 2101–2109, 2010.
5 A. Mohammed and M. Darus, “A new integral operator for meromorphic functions,” Acta Universitatis
Apulensis, no. 24, pp. 231–238, 2010.
6 N. Breaz, D. Breaz, and M. Darus, “Convexity properties for some general integral operators on
uniformly analytic functions classes,” Computers and Mathematics with Applications, vol. 60, no. 12, pp.
3105–3107, 2010.
Advances in
Operations Research
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Mathematical Problems
in Engineering
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Algebra
Hindawi Publishing Corporation
http://www.hindawi.com
Probability and Statistics
Volume 2014
The Scientific
World Journal
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at
http://www.hindawi.com
International Journal of
Advances in
Combinatorics
Hindawi Publishing Corporation
http://www.hindawi.com
Mathematical Physics
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
Hindawi Publishing Corporation
http://www.hindawi.com
Stochastic Analysis
Abstract and
Applied Analysis
Hindawi Publishing Corporation
http://www.hindawi.com
Hindawi Publishing Corporation
http://www.hindawi.com
International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Applied Mathematics
Journal of
Function Spaces
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Optimization
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com
Volume 2014