Paper21-Acta25-2011. 123KB Jun 04 2011 12:03:16 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 25/2011
pp. 235-243

THE EFFECT OF CERTAIN INTEGRAL OPERATORS ON SOME
CLASSES OF ANALYTIC FUNCTIONS

Muhammad Arif, Sarfraz Nawaz Malik and Mohsan Raza
Abstract. The aim of this paper is to introduced subclasses of Janowski functions
with bounded boundary and bounded radius rotations of complex order b and of type ρ.
And also to study the mapping properties of these classes under certain integral operators
defined and studied by Breaz et. al recently.
2000 Mathematics Subject Classification: 30C45, 30C50.
1. Introduction
Let A be the class of functions f (z) of the form
f (z) = z +

∞
X

an z n ,

n=2

which are analytic in the open unit disc E = {z : |z| < 1}. Let Cb (ρ) and Sb∗ (ρ)
be the classes of convex and starlike functions of complex order b (b ∈ C − {0}) and
type ρ (0 ≤ ρ < 1) respectively studied by Frasin [5].
Let P [A, B] be the class of functions h (z), analytic in E with h (0) = 1 and
h (z) ≺

1 + Az
, − 1 ≤ B < A ≤ 1,
1 + Bz

where the symbol ≺ stands for subordination. This class was introduced by Janowski
[6]. It is noted that P [1, −1] ≡ P , where P is the well-known class of functions with
positive real parts. Noor [9] generalized this concept of janowski functions and
defined the class Pk [A, B] as follows.
A function p (z) is said to be in the class Pk [A, B], if and only if,





k 1
k 1
+
h1 (z) −
−
h2 (z) ,
(1.1)
p (z) =
4 2
4 2

235

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

where h1 (z) , h2 (z) ∈ P [A, B]. It is clear that P2 [A, B] ≡ P [A, B] and Pk [1, −1] ≡

Pk , the well-known class given and studied by Pinchuk [13].
The important fact about the class Pk [A, B] is that this class is convex set. That is,
for pi (z) ∈ Pk [A, B] and αi ∈ R with 1 ≤ i ≤ n, we have
n
i=1 αi pi (z)

∈ Pk [A, B].

(1.2)

This can be easily seen from (1.1) by using the fact that the set P [A, B] is convex
[10]. By using all these concepts, we define the following classes.
A function f (z) ∈ A is said to belong to the class Vk [A, B, ρ, b], if and only if,



1
1 zf ′′ (z)
1+
− ρ ∈ Pk [A, B] ,

1−ρ
b f ′ (z)
where −1 ≤ B < A ≤ 1, k ≥ 2, 0 ≤ ρ < 1 and b ∈ C − {0}. When ρ = 0 and b = 1,
we obtain the class Vk [A, B] of janowski functions with bounded boundary rotation,
first discussed by Noor [9].
Similarly, an analytic function f (z) ∈ Rk [A, B, ρ, b], if and only if,




1 zf ′ (z)
1
1+
− 1 − ρ ∈ Pk [A, B] ,
1−ρ
b
f (z)
where −1 ≤ B < A ≤ 1, k ≥ 2, 0 ≤ ρ < 1 and b ∈ C − {0}. When ρ = 0 and
b = 1, we obtain the class Rk [A, B] of functions with bounded radius rotation, first
discussed by Noor [9].

Let us consider the integral operators




f1 (t) α1
fn (t) αn
Fn (z) =z0
···
dt
(1.3)
t
t
and


α

α
Fα1 ...αn (z) =z0 f1′ (t) 1 . . . fn′ (t) n dt,

(1.4)

where fi (z) ∈ A and αi > 0 for all i ∈ {1, 2, . . . , n}.
These operators, given by (1.3) and (1.4), are introduced and studied by Breaz and
Breaz [2] and Breaz et.al [4], respectively. Later on, Breaz and G¨
uney [3] considered
the above integral operators and they obtained their properties on the classes Cb (ρ),
Sb∗ (ρ) of convex and starlike functions of complex order b and type ρ introduced
and studied by Frasin [5]. Recently, Noor [11] discussed the effect of these integral
opeartors on the classes Vk (ρ, b) and Rk (ρ, b) .
In this paper, we investigate some propeties of the above integral operators Fn (z)
and Fα1 ...αn (z) for the classes Vk [A, B, ρ, b] and Rk [A, B, ρ, b] respectively.
In order to derive our main result, we need the following lemmas.
236

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

2. Preliminary Lemmas
Lemma 2.1. Let β, γ, A ∈ C with Re [β + γ] > 0 and let B ∈ [−1, 0] satisfy

either



Re β [1 + ((1 − ρ) A + ρB) B] + γ 1 + B 2 ≥


((1 − ρ) A + ρB) β + βB + B (γ + γ) ,

when B ∈ (−1, 0], or

Reβ [1 + (1 − ρ) A + ρB] > 0 and Re [β [1 − ((1 − ρ) A + ρB)] + 2γ] ≥ 0,
when B = −1. If h (z) = 1 + cn z n + cn+1 z n+1 ... satisfies


1 + {(1 − ρ)A + ρB} z
nzh′ (z)
,
≺
h(z) +
βh(z) + γ

1 + Bz
then
h(z) ≺ Q(z) ≺

1 + {(1 − ρ)A + ρB} z
,
1 + Bz

where
Q(z) =
and

G(z) =
















R1 h 1+Btz i nβ (1−ρ)( BA −1)

t

1
n

R1

dt,

1+Bz

e

(2.2)

γ
1
− ,
βG(z) β

1
n

0

(2.1)

βA
(1−ρ)(t−1)z β+γ
−1

n
n

t

β+γ
−1
n

B 6= 0,

dt,

B = 0.

0

From (2.2), we can deduce the sharp result that h ∈ P (β), with
β = β(ρ, β, γ) = min ReQ(z) = Q(−1).
This result is a special case of one, given in ([7], pp.109).
The following Lemma is a generalization of the result proved in [12].
Lemma 2.2. Let f (z) ∈ Vk [A, B, ρ] . Then, f (z) ∈ Rk [A, B, β], where




A
B (1 − ρ) B
−1 +1
β = β1 (ρ, 1, 0) =
, B 6= 0.
A
(1 − B)(1−ρ)(1− B ) − (1 − B)

(2.3)

Proof. Let
zf ′ (z)
= p(z) =
f (z)



k 1
+
4 2



h1 (z) −

237



k 1
−
4 2



h2 (z) .

(2.4)

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

Logarithmic differentiation of (2.4) yields
(zf ′ (z))′
zp′ (z)
=
p(z)
+
.
f ′ (z)
p(z)
Since f (z) ∈ Vk [A, B, ρ], it follows that
p(z) +

zp′ (z)
∈ Pk [A, B, ρ] .
p(z)

(2.5)

Now, we define
1
φ (z) =
2



z
z
+
(1 − z) (1 − z)2



=

z 1−

z
2
2

(1 − z)




and using (2.4) with convolution technique given by Noor [8], we have

 



k 1
k 1
φ (z)
φ (z)
φ (z)
∗ p (z) =
+
∗ h1 (z) −
−
∗ h2 (z) ,
z
4 2
z
4 2
z
which implies that


 


zp′ (z)
zh′1 (z)
zh′2 (z)
k 1
k 1
p(z) +
=
+
−
h1 (z) +
−
h2 (z) +
. (2.6)
p(z)
4 2
h1 (z)
4 2
h2 (z)
Thus, from (2.5) and (2.6), we have
hi (z) +

zh′i (z)
∈ P [A, B, ρ] ,
hi (z)

i = 1, 2.

We use Lemma 2.1 with −1 ≤ B < A ≤ 1, n = 1, γ = 0, β = 1 > 0, ρ ∈ [0, 1) and
h = hi in (2.1), to have hi ∈ P [A, B, β], where β is given in (2.3) and consequently
p (z) ∈ Pk [A, B, β], which gives the required result. This estimate is best possible,
extremal function Q (z) is given by

(1−ρ)(1− A )
B

 (1+Bz)−(1+Bz)

, if B 6= 0,
A

−1)+1]
Bz [(1−ρ)( B
Q(z) =



 1−e−(1−ρ)(Az) ,
if B = 0.
(1−ρ)Az

238

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

3. Main Results
Theorem 3.1. Let fi (z) ∈ Rk [A, B, ρ, b] for 1 ≤ i ≤ n with −1 ≤ B < A ≤ 1,
0 ≤ ρ < 1, b ∈ C − {0}. Also let αi > 0, 1 ≤ i ≤ n. If
n
i=1 αi

= 1,

then Fn (z) ∈ Vk [A, B, ρ, b].
Proof. From (1.3), we have
zFn′′ (z) n
=
αi
Fn′ (z) i=1
By multiplying (3.1) with




zfi′ (z)
−1 .
fi (z)

(3.1)

1
, we have
b

1 zFn′′ (z) n
1
=i=1 αi
′
b Fn (z)
b



zfi′ (z)
−1
fi (z)



or, equivalently



1 zfi′ (z)
1 zFn′′ (z) n
=
αi 1 +
−1 .
1+
b Fn′ (z) i=1
b fi (z)
Subtracting ρ from both sides of (3.2), we have







1 zfi′ (z)
1 zFn′′ (z)
n
1+
−1
−ρ .
− ρ =i=1 αi 1 +
b Fn′ (z)
b fi (z)
Since fi (z) ∈ Rk [A, B, ρ, b] for 1 ≤ i ≤ n, we have




1 zfi′ (z)
−1
− ρ = (1 − ρ) pi (z) ,
1+
b fi (z)

1 ≤ i ≤ n,

where pi (z) ∈ Pk [A, B]. Using (3.4) in (3.3), we obtain



1 zFn′′ (z)
1+
− ρ = (1 − ρ)ni=1 αi pi (z) .
b Fn′ (z)
Using (1.2), we can have
1
1−ρ




1 zFn′′ (z)
1+
− ρ ∈ Pk [A, B],
b Fn′ (z)

which implies that Fn (z) ∈ Vk [A, B, ρ, b].
239

(3.2)

(3.3)

(3.4)

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

If we take A = 1, B = −1 in Theorem 3.1, we obtain the result proved in [11].
Corollory 3.2. Let fi (z) ∈ Rk (ρ, b) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}.
Also let αi > 0, 1 ≤ i ≤ n. If
n
i=1 αi = 1,
then Fn (z) ∈ Vk (ρ, b).
If k = 2, A = 1, B = −1 in Theorem 3.1, we obtain the result proved in [3].
Corollory 3.3. Let fi (z) ∈ Sb∗ (ρ) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}. Also
let αi > 0, 1 ≤ i ≤ n. If
n
i=1 αi = 1,
then Fn (z) ∈ Cb (ρ).
Theorem 3.4. Let fi (z) ∈ Vk [A, B, ρ, 1] for 1 ≤ i ≤ n with −1 ≤ B < A ≤ 1,
B 6= 0, 0 ≤ ρ < 1. Also let αi > 0, 1 ≤ i ≤ n. If
n
i=1 αi

= 1,

then Fn (z) ∈ Vk [A, B, β, 1], where β is given by (2.3).
Proof. From (3.2) with b = 1, we have

 ′ 

zFn′′ (z)
zfi (z)
n
1+ ′
=i=1 αi
Fn (z)
fi (z)
or, equivalently




 ′
zFn′′ (z)
zfi (z)
n
1+ ′
−β .
− β =i=1 αi
Fn (z)
fi (z)

(3.5)

Since fi (z) ∈ Vk [A, B, ρ, 1] for 1 ≤ i ≤ n, then by using Lemma 2.2, we have
fi (z) ∈ Rk [A, B, β, 1], where β is given by (2.3). That is,
zfi′ (z)
− β = (1 − β) pi (z) ,
fi (z)

1 ≤ i ≤ n,

where pi (z) ∈ Pk [A, B]. Using (3.6) in (3.5), we obtain



zFn′′ (z)
1+ ′
− β = (1 − β)ni=1 αi pi (z) .
Fn (z)
Using (1.2), we can have
1
1−β




zFn′′ (z)
1+ ′
− β ∈ Pk [A, B],
Fn (z)
240

(3.6)

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

which implies that Fn (z) ∈ Vk [A, B, β, 1].
Set n = 1 with α1 = 1, in Theorem 3.4, we obtain.
Corollory 3.5. Let f (z) ∈ Vk [A, B, ρ] for −1 ≤ B < A ≤ 1, B 6= 0. Then
the Alexandar operator F1 (z), defined in [1], belongs to the class Vk [A, B, β] for
−1 ≤ B < A ≤ 1, B 6= 0, where β is given by (2.3) .
For A = 1, B = −1, ρ = 0 and k = 2 in Corollary 3.5, we have the well known result,
that is,
 
1
f (z) ∈ C (0) ⇒ F1 (z) ∈ C
.
2
By setting A = 1, B = −1 in Theorem 3.4, we obtain the following result.
Corollory 3.6. Let fi (z) ∈ Vk (ρ, 1) for 1 ≤ i ≤ n with 0 ≤ ρ < 1. Also let αi > 0,
1 ≤ i ≤ n. If
n
i=1 αi = 1,
then Fn (z) ∈ Vk (β, 1) ,where β is given by (2.3).
The above result in Corollary 3.6 is special case of the results proved in [11].
Theorem 3.7. Let fi (z) ∈ Vk [A, B, ρ, b] for 1 ≤ i ≤ n with −1 ≤ B < A ≤ 1,
0 ≤ ρ < 1, b ∈ C − {0}. Also let αi > 0, 1 ≤ i ≤ n. If
n
i=1 αi

= 1,

then Fα1 ...αn (z) ∈ Vk [A, B, ρ, b].
Proof. From (1.4), we have
Fα′′1 ...αn (z) n
=
αi
Fα′ 1 ...αn (z) i=1
By multiplying both sides with



fi′′ (z)
fi′ (z)



.

z
, we have
b

1 zFα′′1 ...αn (z) n
1
=
αi
b Fα′ 1 ...αn (z) i=1 b



zfi′′ (z)
fi′ (z)



This relation is equivalent to






1 zfi′′ (z)
1 zFα′′1 ...αn (z)
n
1+
−ρ .
− ρ =i=1 αi 1 +
b Fα′ 1 ...αn (z)
b fi′ (z)
Since fi (z) ∈ Vk [A, B, ρ, b] for 1 ≤ i ≤ n, we have


1 zfi′′ (z)
− ρ = (1 − ρ) pi (z) ,
1+
b fi′ (z)
241

1 ≤ i ≤ n,

(3.7)

(3.8)

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

where pi (z) ∈ Pk [A, B]. Using (3.8) in (3.7), we obtain



1 zFα′′1 ...αn (z)
1+
− ρ = (1 − ρ)ni=1 αi pi (z) .
b Fα′ 1 ...αn (z)
Using the fact given in (1.2), we get



1 zFα′′1 ...αn (z)
1
1+
− ρ ∈ Pk [A, B].
1−ρ
b Fα′ 1 ...αn (z)
This implies that Fα1 ...αn (z) ∈ Vk [A, B, ρ, b].
When A = 1, B = −1 in Theorem 3.7, we obtain the result proved in [11].
Corollory 3.8. Let fi (z) ∈ Vk (ρ, b) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}.
Also let αi > 0, 1 ≤ i ≤ n. If
n
i=1 αi = 1,
then Fα1 ...αn (z) ∈ Vk (ρ, b).
If k = 2, A = 1, B = −1 in Theorem 3.7, we have the result discussed in [3].
Corollory 3.9. Let fi (z) ∈ Sb∗ (ρ) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}. Also
let αi > 0, 1 ≤ i ≤ n. If
n
i=1 αi = 1,
then Fα1 ...αn (z) ∈ Cb (ρ).
Acknowlogment: The principal author would like to express gratitude to Professor
Dr. Ihsan Ali, Vice Chancellor Adul Wali Khan Univaersity Mardan for his support
and providing excellent research facilities.

References
[1]

I.W. Alexander, Functions which map the interior of the unit circle upon simple
regions, Ann. of Math., 17(1915), 12-22.

[2]

D. Breaz, N. Breaz, Two integral operator, Studia Universitatis Babes-Bolyai,
Mathematica, Clunj-Napoca, 3(2002), 13-21.

[3]

¨ Guney, The integral operator on the classes S ∗ (b) and Cα (b),
D. Breaz, H. O
α
J. Math. Ineq. vol. 2, 1(2008), 97-100.

[4]

D. Breaz, S. Owa, N. Breaz, A new integral univalent operator, Acta Univ.
Apulensis Math. Inform., 16(2008), 11-16.

242

M. Arif, S. N. Malik and M. Raza -The Effect of Certain Integral Operators...

[5]

B. A. Frasin, Family of analytic functions of complex order, Acta Math. Acad.
Paed. Ny. 22(2006), 179-191.

[6]

W. Janowski, Some extremal problems for certain families of analytic functions,
Ann. Polon. Math., 28 (1973), 297-326.

[7]

S. S. Miller, P. T. Mocanu,Differential Subordinations, Theory and Applications, Marcel Dekker, Inc., New York, Basel, 2000.

[8]

K. I. Noor, Classes of analytic functions defined by Hadamard product, New
Zealand J. Math. 24 (1995), 53-64.

[9]

K. I. Noor, On some integral operators for certain families of analytic function,
Tamkang J. Math., 22(1991), 113-117.

[10] K. I. Noor, On some univalent integral operators, J. Math. Anal. and Appl.,
128 (1987), 586–592.
[11] K.I.Noor, M.Arif, W. Ul-Haq, Some properties of certain integral operators,
Acta Univ. Apulensis Math. Inform., 21(2010), 89-95.
[12] K. I. Noor, W. Haq, M. Arif and S. Mustafa, On bounded boundary and bounded
radius rotations, J. Inequ. Appl., vol. (2009) art. ID 813687, pp 12.
[13] B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math., 10(1971),
7-16.
Muhammad Arif
Department of Mathematics
Abdul Wali Khan University Mardan, Pakistan.
E-mail: marifmaths@yahoo.com
Sarfraz Nawaz Malik
Department of Mathematics
COMSATS Institute of Information Technology, Islamabad Pakistan.
E-mail: snmalik110@yahoo.com
Mohsan Raza
Department of Mathematics
COMSATS Institute of Information Technology, Islamabad Pakistan.
E-mail: mohsan976@yahoo.com

243