09-21.2010. 98KB Jun 04 2011 12:07:16 AM
Acta Universitatis Apulensis
ISSN: 1582-5329
No. 21/2010
pp. 89-95
SOME PROPERTIES OF CERTAIN INTEGRAL OPERATORS
Khalida Inayat Noor, Muhammad Arif and Wasim Ul Haq
Abstract. In this paper, we consider certain subclasses of analytic functions with
bounded radius and bounded boundary rotation and study the mapping properties of these
classes under certain integral operators introduced by Breaz et. al recently.
Key Words: Bounded boundary and bounded radius rotations, Integral operators.
2000 Mathematics Subject Classification: 30C45, 30C50.
1. Introduction
Let A be the class of all 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 [3].
Let Pk (ρ) be the class of functions p(z) analytic in E with p(0) = 1 and
Z2π
Re p(z) − ρ
dθ ≤ kπ, z = reiθ ,
1−ρ
0
where k ≥ 2 and 0 ≤ ρ < 1. We note that, for ρ = 0, we obtain the class Pk
defined and studied in [10], and for ρ = 0, k = 2, we have the well known class P of
functions with positive real part. The case k = 2 gives the class P (ρ) of functions
with positive real part greater than ρ.
A function f (z) ∈ A is said to belong to the class Vk (ρ, b) if and only if
1+
1 zf ′′ (z)
∈ Pk (ρ),
b f ′ (z)
89
(1.1)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
where k ≥ 2, 0 ≤ ρ < 1 and b ∈ C − {0}. When ρ = 0 and b = 1, we obtain the class
Vk of functions with bounded boundary rotation, first discussed by Paatero [2].
Similarly, an analytic function f (z) ∈ Rk (ρ, b) if and only if
1 zf ′ (z)
1+
− 1 ∈ Pk (ρ)
(1.2)
b
f (z)
where k ≥ 2, 0 ≤ ρ < 1 and b ∈ C − {0}. When ρ = 0 and b = 1, we obtain the class
Rk of functions with bounded radius rotation [2]. For more details see [6, 7, 8]. Let
us consider the integral operators
Fn (z) =
Zz
f1 (t)
t
α 1
···
fn (t)
t
α n
dt
(1.3)
0
and
Fα1 ...αn (z) =
Zz
0
′ α1
α
f1 (t)
. . . 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. Recently, 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].
In this paper, we investigate some propeties of the above integral operators Fn (z)
and Fα1 ...αn (z) for the classes Vk (ρ, b) and Rk (ρ, b) respectively. In order to derive
our main result, we need the following lemma.
Lemma 1.1 . [9] Let f (z) ∈ Vk (α), 0 ≤ α < 1. Then f (z) ∈ Rk (β), where
β=
i
p
1h
(2α − 1) + 4α2 − 4α + 9 .
4
(1.5)
2. Main Results
Theorem 2.1. 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
0 ≤ (ρ − 1)
n
X
i=1
90
αi + 1 < 1,
(2.1)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
then Fn (z) ∈ Vk (λ, b) with
λ = (ρ − 1)
n
X
αi + 1.
(2.2)
i=1
Proof. From (1.3), we have
n
zFn′′ (z) X
=
αi
Fn′ (z)
i=1
zfi′ (z)
−1 .
fi (z)
(2.3)
1
, we have
b
n
1 zFn′′ (z) X 1 zfi′ (z)
−
1
=
α
i
b Fn′ (z)
b fi (z)
Then by multiplying (2.3) with
i=1
=
n
X
i=1
X
n
1 zfi′ (z)
−1 −
αi 1 +
αi
b fi (z)
i=1
or, equivalently
n
n
X
X
1 zfi′ (z)
1 zFn′′ (z)
−1 .
=1−
αi +
αi 1 +
1+
b Fn′ (z)
b fi (z)
i=1
(2.4)
i=1
Subtracting and adding ρ on the right hand side of (2.4), we have
X
n
1 zFn′′ (z)
1 zfi′ (z)
1+
−1
−ρ ,
−λ =
αi 1 +
b Fn′ (z)
b fi (z)
(2.5)
i=1
where λ is given by (2.2). Taking real part of (2.5) and then simple computation
gives
Z2π
Z2π
n
′′ (z)
X
zF
1
1 zfi′ (z)
n
dθ.
dθ ≤
Re 1 +
−
1
−
ρ
−
λ
α
Re
1
+
i
′
b Fn (z)
b fi (z)
i=1
0
0
(2.6)
Since fi (z) ∈ Rk (ρ, b) for 1 ≤ i ≤ n, we have
′
Z2π
dθ ≤ (1 − ρ) kπ.
Re 1 + 1 zfi (z) − 1
−
ρ
b fi (z)
0
91
(2.7)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
Using (2.7) in (2.6), we obtain
Z2π
′′
Re 1 + 1 zFn (z) − λ dθ ≤ (1 − λ) kπ.
b Fn′ (z)
0
Hence Fn (z) ∈ Vk (λ, b) with λ is given by (2.2).By setting k = 2 in Theorem 2.1,
we obtain the following result proved in [3].
Corollary 2.2. Let fi (z) ∈ Sb∗ (ρ) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}.
Also let αi > 0, 1 ≤ i ≤ n. If
0 ≤ (ρ − 1)
n
X
αi + 1 < 1,
i=1
then Fn (z) ∈ Cb (λ) with λ = (ρ − 1)
n
X
αi + 1.
i=1
Theorem 2.3. Let fi (z) ∈ Vk (ρ, 1) for 1 ≤ i ≤ n with 0 ≤ ρ < 1. Also let
αi > 0, 1 ≤ i ≤ n. If
n
X
0 ≤ (β − 1)
αi + 1 < 1,
i=1
then Fn (z) ∈ Vk (λ, 1) with λ = (β − 1)
n
X
αi + 1 and β is given by (1.5).
i=1
Proof. From (2.6) with b = 1, we have
Z2π
Z2π ′
n
′′ (z)
X
zF
zfi (z)
Re 1 + n
dθ.
−
β
−
λ
dθ
≤
α
Re
i
Fn′ (z)
fi (z)
i=1
0
(2.8)
0
Since fi (z) ∈ Vk (ρ, 1) for 1 ≤ i ≤ n, then by using Lemma 1.1, we have
Z2π ′
Re zfi (z) − β dθ ≤ (1 − β) kπ,
fi (z)
0
where β is given by (1.5) with α = ρ. Using (2.9) in (2.8), we obtain
Z2π
′′
Re 1 + zFn (z) − λ dθ ≤ (1 − λ) kπ.
Fn′ (z)
0
92
(2.9)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
Hence Fn (z) ∈ Vk (λ, b) with λ = (β − 1)
n
X
αi + 1.
i=1
Set n = 1, α1 = 1, α2 = · · · = αn = 0 in Theorem 2.3, we obtain.
Corollary 2.4. Let f (z) ∈ Vk (ρ). Then the Alexandar operator F0 (z), defined
in [1] , belongs to the class Vk (β), where β is given by (1.5).
For ρ = 0 and k = 2 in Corollary 2.4, we have the well known result, that is,
1
f (z) ∈ C (0) implies F0 (z) ∈ C
.
2
Theorem 2.5. 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
0 ≤ (ρ − 1)
n
X
αi + 1 < 1,
i=1
then Fα1 ...αn (z) ∈ Vk (µ, b) with µ = (ρ − 1)
n
X
αi + 1.
i=1
Proof. From (1.4), we have
n
Fα′′1 ...αn (z) X
=
αi
Fα′ 1 ...αn (z)
i=1
fi′′ (z)
fi′ (z)
.
z
, we have
b
n
1 zFα′′1 ...αn (z) X 1 zfi′′ (z)
=
αi
b Fα′ 1 ...αn (z)
b
fi′ (z)
Then by multiplying both sides with
i=1
This relation is equivalent to
X
n
1 zfi′′ (z)
1 zFα′′1 ...αn (z)
−ρ ,
−µ =
αi 1 +
1+
b Fα′ 1 ...αn (z)
b fi′ (z)
(2.10)
i=1
where µ = (ρ − 1)
n
X
αi +1. Taking real part of (2.10) and then simple computation
i=1
gives us
Z2π
Z2π
n
′′
X
1 zfi′′ (z)
Re 1 + 1 zFα1 ...αn (z) − µ dθ ≤
dθ.
−
ρ
α
Re
1
+
i
b Fα′ 1 ...αn (z)
b fi′ (z)
0
i=1
0
(2.11)
93
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
Since fi (z) ∈ Vk (ρ, b) for 1 ≤ i ≤ n, we have
Z2π
′′
Re 1 + 1 zfi (z) − ρ dθ ≤ (1 − ρ) kπ.
′
b fi (z)
(2.12)
0
Using (2.12) in (2.11), we obtain
Z2π
′′
Re 1 + 1 zFα1 ...αn (z) − µ dθ ≤ (1 − µ) kπ.
′
b Fα1 ...αn (z)
0
Hence Fα1 ...αn (z) ∈ Vk (µ, b) with µ = (ρ − 1)
n
X
αi + 1.
i=1
By setting k = 2 in Theorem 2.5, we obtain the following result proved in [3].
Corollary 2.6. Let fi (z) ∈ Cb (ρ) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}.
Also let αi > 0, 1 ≤ i ≤ n. If
0 ≤ (ρ − 1)
n
X
αi + 1 < 1,
i=1
then Fα1 ...αn (z) ∈ Cb (µ) with µ = (ρ − 1)
n
X
αi + 1.
i=1
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.
¨ Guney, The integral operator on the classes S ∗ (b) and Cα (b),
[3] 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.
[5] B. A. Frasin, Family of analytic functions of complex order, Acta Math. Acad.
Paed. Ny. 22(2006), 179-191.
[6] K. I. Noor, Properties of certain analytic functions, J. Natural Geometry,
7(1995), 11-20.
94
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
[7] K. I. Noor, On some subclassess of fuctions with bounded boundary and
bounded radius rotation, Pan. Amer. Math. J., 6(1996), 75-81.
[8] K. I. Noor, On analytic functions related to certain family of integral operators,
J. Ineq. Pure and Appl. Math. 7(2)(26)(2006).
[9] K.I.Noor, W. Ul-Haq, M.Arif, S. Mustafa, On Bounded Boundary and Bounded
Radius Rotations, J. Inequ. Appl., vol. (2009) articls ID 813687, pages 12.
[10] B. Pinchuk, Functions with bounded boundary rotation, Isr. J. Math., 10(1971),
7-16.
Prof. Dr. Khalida Inayat Noor
Department of Mathematics
COMSATS Institute of Information Technology,
Islamabad Pakistan
email: [email protected]
Dr. Muhammad Arif
Department of Mathematics
Abdul Wali Khan University Mardan
Pakistan
email: [email protected]
95
ISSN: 1582-5329
No. 21/2010
pp. 89-95
SOME PROPERTIES OF CERTAIN INTEGRAL OPERATORS
Khalida Inayat Noor, Muhammad Arif and Wasim Ul Haq
Abstract. In this paper, we consider certain subclasses of analytic functions with
bounded radius and bounded boundary rotation and study the mapping properties of these
classes under certain integral operators introduced by Breaz et. al recently.
Key Words: Bounded boundary and bounded radius rotations, Integral operators.
2000 Mathematics Subject Classification: 30C45, 30C50.
1. Introduction
Let A be the class of all 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 [3].
Let Pk (ρ) be the class of functions p(z) analytic in E with p(0) = 1 and
Z2π
Re p(z) − ρ
dθ ≤ kπ, z = reiθ ,
1−ρ
0
where k ≥ 2 and 0 ≤ ρ < 1. We note that, for ρ = 0, we obtain the class Pk
defined and studied in [10], and for ρ = 0, k = 2, we have the well known class P of
functions with positive real part. The case k = 2 gives the class P (ρ) of functions
with positive real part greater than ρ.
A function f (z) ∈ A is said to belong to the class Vk (ρ, b) if and only if
1+
1 zf ′′ (z)
∈ Pk (ρ),
b f ′ (z)
89
(1.1)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
where k ≥ 2, 0 ≤ ρ < 1 and b ∈ C − {0}. When ρ = 0 and b = 1, we obtain the class
Vk of functions with bounded boundary rotation, first discussed by Paatero [2].
Similarly, an analytic function f (z) ∈ Rk (ρ, b) if and only if
1 zf ′ (z)
1+
− 1 ∈ Pk (ρ)
(1.2)
b
f (z)
where k ≥ 2, 0 ≤ ρ < 1 and b ∈ C − {0}. When ρ = 0 and b = 1, we obtain the class
Rk of functions with bounded radius rotation [2]. For more details see [6, 7, 8]. Let
us consider the integral operators
Fn (z) =
Zz
f1 (t)
t
α 1
···
fn (t)
t
α n
dt
(1.3)
0
and
Fα1 ...αn (z) =
Zz
0
′ α1
α
f1 (t)
. . . 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. Recently, 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].
In this paper, we investigate some propeties of the above integral operators Fn (z)
and Fα1 ...αn (z) for the classes Vk (ρ, b) and Rk (ρ, b) respectively. In order to derive
our main result, we need the following lemma.
Lemma 1.1 . [9] Let f (z) ∈ Vk (α), 0 ≤ α < 1. Then f (z) ∈ Rk (β), where
β=
i
p
1h
(2α − 1) + 4α2 − 4α + 9 .
4
(1.5)
2. Main Results
Theorem 2.1. 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
0 ≤ (ρ − 1)
n
X
i=1
90
αi + 1 < 1,
(2.1)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
then Fn (z) ∈ Vk (λ, b) with
λ = (ρ − 1)
n
X
αi + 1.
(2.2)
i=1
Proof. From (1.3), we have
n
zFn′′ (z) X
=
αi
Fn′ (z)
i=1
zfi′ (z)
−1 .
fi (z)
(2.3)
1
, we have
b
n
1 zFn′′ (z) X 1 zfi′ (z)
−
1
=
α
i
b Fn′ (z)
b fi (z)
Then by multiplying (2.3) with
i=1
=
n
X
i=1
X
n
1 zfi′ (z)
−1 −
αi 1 +
αi
b fi (z)
i=1
or, equivalently
n
n
X
X
1 zfi′ (z)
1 zFn′′ (z)
−1 .
=1−
αi +
αi 1 +
1+
b Fn′ (z)
b fi (z)
i=1
(2.4)
i=1
Subtracting and adding ρ on the right hand side of (2.4), we have
X
n
1 zFn′′ (z)
1 zfi′ (z)
1+
−1
−ρ ,
−λ =
αi 1 +
b Fn′ (z)
b fi (z)
(2.5)
i=1
where λ is given by (2.2). Taking real part of (2.5) and then simple computation
gives
Z2π
Z2π
n
′′ (z)
X
zF
1
1 zfi′ (z)
n
dθ.
dθ ≤
Re 1 +
−
1
−
ρ
−
λ
α
Re
1
+
i
′
b Fn (z)
b fi (z)
i=1
0
0
(2.6)
Since fi (z) ∈ Rk (ρ, b) for 1 ≤ i ≤ n, we have
′
Z2π
dθ ≤ (1 − ρ) kπ.
Re 1 + 1 zfi (z) − 1
−
ρ
b fi (z)
0
91
(2.7)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
Using (2.7) in (2.6), we obtain
Z2π
′′
Re 1 + 1 zFn (z) − λ dθ ≤ (1 − λ) kπ.
b Fn′ (z)
0
Hence Fn (z) ∈ Vk (λ, b) with λ is given by (2.2).By setting k = 2 in Theorem 2.1,
we obtain the following result proved in [3].
Corollary 2.2. Let fi (z) ∈ Sb∗ (ρ) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}.
Also let αi > 0, 1 ≤ i ≤ n. If
0 ≤ (ρ − 1)
n
X
αi + 1 < 1,
i=1
then Fn (z) ∈ Cb (λ) with λ = (ρ − 1)
n
X
αi + 1.
i=1
Theorem 2.3. Let fi (z) ∈ Vk (ρ, 1) for 1 ≤ i ≤ n with 0 ≤ ρ < 1. Also let
αi > 0, 1 ≤ i ≤ n. If
n
X
0 ≤ (β − 1)
αi + 1 < 1,
i=1
then Fn (z) ∈ Vk (λ, 1) with λ = (β − 1)
n
X
αi + 1 and β is given by (1.5).
i=1
Proof. From (2.6) with b = 1, we have
Z2π
Z2π ′
n
′′ (z)
X
zF
zfi (z)
Re 1 + n
dθ.
−
β
−
λ
dθ
≤
α
Re
i
Fn′ (z)
fi (z)
i=1
0
(2.8)
0
Since fi (z) ∈ Vk (ρ, 1) for 1 ≤ i ≤ n, then by using Lemma 1.1, we have
Z2π ′
Re zfi (z) − β dθ ≤ (1 − β) kπ,
fi (z)
0
where β is given by (1.5) with α = ρ. Using (2.9) in (2.8), we obtain
Z2π
′′
Re 1 + zFn (z) − λ dθ ≤ (1 − λ) kπ.
Fn′ (z)
0
92
(2.9)
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
Hence Fn (z) ∈ Vk (λ, b) with λ = (β − 1)
n
X
αi + 1.
i=1
Set n = 1, α1 = 1, α2 = · · · = αn = 0 in Theorem 2.3, we obtain.
Corollary 2.4. Let f (z) ∈ Vk (ρ). Then the Alexandar operator F0 (z), defined
in [1] , belongs to the class Vk (β), where β is given by (1.5).
For ρ = 0 and k = 2 in Corollary 2.4, we have the well known result, that is,
1
f (z) ∈ C (0) implies F0 (z) ∈ C
.
2
Theorem 2.5. 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
0 ≤ (ρ − 1)
n
X
αi + 1 < 1,
i=1
then Fα1 ...αn (z) ∈ Vk (µ, b) with µ = (ρ − 1)
n
X
αi + 1.
i=1
Proof. From (1.4), we have
n
Fα′′1 ...αn (z) X
=
αi
Fα′ 1 ...αn (z)
i=1
fi′′ (z)
fi′ (z)
.
z
, we have
b
n
1 zFα′′1 ...αn (z) X 1 zfi′′ (z)
=
αi
b Fα′ 1 ...αn (z)
b
fi′ (z)
Then by multiplying both sides with
i=1
This relation is equivalent to
X
n
1 zfi′′ (z)
1 zFα′′1 ...αn (z)
−ρ ,
−µ =
αi 1 +
1+
b Fα′ 1 ...αn (z)
b fi′ (z)
(2.10)
i=1
where µ = (ρ − 1)
n
X
αi +1. Taking real part of (2.10) and then simple computation
i=1
gives us
Z2π
Z2π
n
′′
X
1 zfi′′ (z)
Re 1 + 1 zFα1 ...αn (z) − µ dθ ≤
dθ.
−
ρ
α
Re
1
+
i
b Fα′ 1 ...αn (z)
b fi′ (z)
0
i=1
0
(2.11)
93
K.I. Noor, M. Arif, W. Ul Haq - Some properties of certain integral operators
Since fi (z) ∈ Vk (ρ, b) for 1 ≤ i ≤ n, we have
Z2π
′′
Re 1 + 1 zfi (z) − ρ dθ ≤ (1 − ρ) kπ.
′
b fi (z)
(2.12)
0
Using (2.12) in (2.11), we obtain
Z2π
′′
Re 1 + 1 zFα1 ...αn (z) − µ dθ ≤ (1 − µ) kπ.
′
b Fα1 ...αn (z)
0
Hence Fα1 ...αn (z) ∈ Vk (µ, b) with µ = (ρ − 1)
n
X
αi + 1.
i=1
By setting k = 2 in Theorem 2.5, we obtain the following result proved in [3].
Corollary 2.6. Let fi (z) ∈ Cb (ρ) for 1 ≤ i ≤ n with 0 ≤ ρ < 1, b ∈ C − {0}.
Also let αi > 0, 1 ≤ i ≤ n. If
0 ≤ (ρ − 1)
n
X
αi + 1 < 1,
i=1
then Fα1 ...αn (z) ∈ Cb (µ) with µ = (ρ − 1)
n
X
αi + 1.
i=1
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,
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Prof. Dr. Khalida Inayat Noor
Department of Mathematics
COMSATS Institute of Information Technology,
Islamabad Pakistan
email: [email protected]
Dr. Muhammad Arif
Department of Mathematics
Abdul Wali Khan University Mardan
Pakistan
email: [email protected]
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