Journal of Inequalities in Pure and
Applied Mathematics
ON ANALYTIC FUNCTIONS RELATED TO CERTAIN
FAMILY OF INTEGRAL OPERATORS
volume 7, issue 2, article 69,
2006.
KHALIDA INAYAT NOOR
Mathematics Department
COMSATS Institute of Information Techonolgy
Islamabad, Pakistan
Received 02 December, 2005;
accepted 11 January, 2006.
Communicated by: N.E. Cho
EMail: khalidanoor@hotmail.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
354-05
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Abstract
P
n
Let A be the class of functions f(z) = z + ∞
n=2 an z . . . , analytic in the open
unit disc E. A certain integral operator is used to define some subclasses of A
and their inclusion properties are studied.
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words: Convex and starlike functions of order α, Quasi-convex functions, Integral operator.
On Analytic Functions Related
to Certain Family of Integral
Operators
This research is supported by the Higher Education Commission, Pakistan, through
grant No: 1-28/HEC/HRD/2005/90.
Khalida Inayat Noor
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
8
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1.
Introduction
Let A denote the class of functions
(1.1)
f (z) = z +
∞
X
an z n ,
n=2
which are analytic in the open disk E = {z : |z| < 1}. Let the functions fi be
defined for i = 1, 2, by
(1.2)
fi (z) = z +
∞
X
On Analytic Functions Related
to Certain Family of Integral
Operators
an,i z n .
n=2
Khalida Inayat Noor
The modified Hadamard product (convolution) of f1 and f2 is defined here by
(f1 ? f2 )(z) = z +
∞
X
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n
an,1 an,2 z .
Contents
n=2
Let Pk (β) be the class of functions h(z) analytic in the unit disc E satisfying
the properties h(0) = 1 and
Z 2π h(z) − β (1.3)
Re 1 − β dθ ≤ kπ,
0
where z = reiθ , k ≥ 2 and 0 ≤ β < 1, see [4]. For β = 0, we obtain the
class Pk defined by Pinchuk [5]. The case k = 2, β = 0 gives us the class P
of functions with positive real part, and k = 2, P2 (β) = P (β) is the class of
functions with positive real part greater than β.
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Also we can write for h ∈ Pk (β)
Z
1 2π 1 + (1 − 2β)ze−it
(1.4)
h(z) =
dµ(t),
2 0
1 − ze−it
where µ(t) is a function with bounded variation on [0, 2π] such that
Z 2π
Z 2π
(1.5)
dµ(t) = 2 and
|dµ(t)| ≤ k.
0
0
From (1.4) and (1.5), we can write, for h ∈ Pk (β),
k 1
k 1
(1.6)
h(z) =
+
h1 (z) −
−
h2 (z),
4 2
4 2
We have the following classes:
Rk (α) = f : f ∈ A and
zf 0 (z)
∈ Pk (α),
f (z)
h1 , h2 ∈ P (β).
z ∈ E,
Khalida Inayat Noor
0≤α<1 .
We note that R2 (α) = S ? (α) is the class of starlike functions of order α.
(zf 0 (z))0
Vk (α) = f : f ∈ A and
∈ Pk (α), z ∈ E, 0 ≤ α < 1 .
f 0 (z)
Note that V2 (α) = C(α) is the class of convex functions of order α.
Tk (β, α) = f : f ∈ A, g ∈ R2 (α)
and
zf 0 (z)
∈ Pk (β),
g(z)
z ∈ E,
On Analytic Functions Related
to Certain Family of Integral
Operators
0 ≤ α, β < 1 .
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We note that T2 (0, 0) is the class K of close-to-convex univalent functions.
Tk? (β, α)
= f : f ∈ A, g ∈ V2 (α) and
(zf 0 (z))0
∈ Pk (β),
g 0 (z)
z ∈ E,
0 ≤ α, β < 1 .
In particular, the class T2? (β, α) = C ? (β, α) was considered by Noor [3] and for
T2? (0, 0) = C ? is the class of quasi-convex univalent functions which was first
introduced and studied in [2].
It can be easily seen from the above definitions that
(1.7)
f (z) ∈ Vk (α)
⇐⇒
On Analytic Functions Related
to Certain Family of Integral
Operators
Khalida Inayat Noor
0
zf (z) ∈ Rk (α)
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and
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(1.8)
f (z) ∈ Tk? (β, α)
⇐⇒
zf 0 (z) ∈ Tk (β, α).
We consider the following integral operator Lµλ : A −→ A, for λ > −1; µ > 0;
f ∈ A,
µ−1
Z z
t
λ+µ µ
µ
λ−1
Lλ f (z) = Cλ
t
1−
f (t)dt
zλ 0
z
∞
Γ(λ + µ + 1) X Γ(λ + n)
=z+
(1.9)
an z n ,
Γ(λ + 1) n=2 Γ(λ + µ + n)
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where Γ denotes the Gamma function. From (1.9), we can obtain the wellknown generalized Bernadi operator as follows:
Z
µ + 1 z µ−1
Iµ f (z) =
t f (t)dt
zµ
0
∞
X
µ+1
=z+
an z n , µ > −1; f ∈ A.
µ
+
n
n=2
We now define the following subclasses of A by using the integral operator
Lµλ .
On Analytic Functions Related
to Certain Family of Integral
Operators
Definition 1.1. Let f ∈ A. Then f ∈ Rk (λ, µ, α) if and only if
for z ∈ E.
Lµλ f ∈ Rk (α),
Khalida Inayat Noor
Definition 1.2. Let f ∈ A. Then f ∈ Vk (λ, µ, α) if and only if
for z ∈ E.
Lµλ f ∈ Vk (α),
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Definition 1.3. Let f ∈ A. Then f ∈ Tk (λ, µ, β, α) if and only if
Tk (β, α), for z ∈ E.
Lµλ f
Definition 1.4. Let f ∈ A. Then f ∈ Tk? (λ, µ, β, α) if and only if
Tk? (β, α), for z ∈ E.
Lµλ f ∈
∈
We shall need the following result.
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Lemma 1.1 ([1]). Let u = u1 + iu2 and v = v1 + iv2 and let Φ be a complexvalued function satisfying the conditions:
(i) Φ(u, v) is continuous in a domain D ⊂ C2 ,
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(ii) (1, 0) ∈ D and Φ(1, 0) > 0.
(iii) Re Φ(iu2 , v1 ) ≤ 0, whenever (iu2 , v1 ) ∈ D and v1 ≤ − 12 (1 + u22 ).
P
m
0
If h(z) = 1 + ∞
m=2 cm z is a function analytic in E such that (h(z), zh (z)) ∈
D and Re Φ(h(z), zh0 (z)) > 0 for z ∈ E, then Re h(z) > 0 in E.
On Analytic Functions Related
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Operators
Khalida Inayat Noor
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2.
Main Results
Theorem 2.1. Let f ∈ A, λ > −1, µ > 0 and λ + µ > 0. Then Rk (λ, µ, 0) ⊂
Rk (λ, µ + 1, α), where
(2.1)
α=
2
p
,
(β + 1) + β 2 + 2β + 9
with β = 2(λ + µ).
Proof. Let f ∈ Rk (λ, µ, 0) and let
0
zLµ+1
k 1
k 1
λ f (z)
= p(z) =
+
p1 (z) −
−
p2 (z),
4
2
4
2
Lµ+1
f
(z)
λ
where p(0) = 1 and p(z) is analytic in E. From (1.9), it can easily be seen that
0
µ
µ+1
(2.2)
z Lµ+1
λ f (z) = (λ + µ + 1)Lλ f (z) − (λ + µ)Lλ f (z).
On Analytic Functions Related
to Certain Family of Integral
Operators
Khalida Inayat Noor
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Some computation and use of (2.2) yields
z (Lµλ f (z))0
zp0 (z)
= p(z) +
∈ Pk ,
Lµλ f (z)
p(z) + λ + µ
z ∈ E.
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Let
∞
X
(λ + µ) + j
zj
λ
+
µ
+
1
j=1
λ+µ
z
1
z
=
+
.
λ+µ+1 1−z
λ + µ + 1 (1 − z)2
Φλ,µ (z) =
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Then
p(z) ? Φλ,µ (z)
zp0 (z)
= p(z) +
p(z) + λ + µ
k 1
k 1
=
+
[p1 (z) ? Φλ,µ (z)] −
−
[p2 (z) ? Φλ,µ (z)]
4 2
4 2
k 1
k 1
zp01 (z)
zp02 (z)
=
+
p1 (z) +
−
−
p2 (z) +
,
4 2
p1 (z) + λ + µ
4 2
p2 (z) + λ + µ
and this implies that
Khalida Inayat Noor
pi (z) +
zp0i (z)
pi (z) + λ + µ
∈ P,
z ∈ E.
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We want to show that pi (z) ∈ P (α), where α is given by (2.1) and this will
show that p ∈ Pk (α) for z ∈ E. Let
pi (z) = (1 − α)hi (z) + α,
Then
On Analytic Functions Related
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i = 1, 2.
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(1 − α)hi (z) + α +
α)zh0i (z)
(1 −
(1 − α)hi (z) + α + λ + µ
We form the functional Ψ(u, v) by choosing u = hi (z),
Ψ(u, v) = (1 − α)u + α +
∈ P.
v = zh0i . Thus
(1 − α)v
.
(1 − α)u + (α + λ + µ)
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The first two conditions of Lemma 1.1 are clearly satisfied. We verify the condition (iii) as follows.
Re Ψ(iu2 , v1 ) = α +
(1 − α)(α + λ + µ)v1
2.
(α + λ + µ)2 + (1 − α)2 u2
2
By putting v1 ≤ −
(1+u2 )
,
2
we obtain
Re Ψ(iu2 , v1 )
1 (1 − α)(α + λ + µ)(1 + u22 )
≤α−
2 (α + λ + µ)2 + (1 − α)2 u22
2
2α(α + λ + µ)2 + 2α(1 − α)2 u2 − (1 − α)(α + λ + µ) − (1 − α)(α + λ + µ)u22
=
2[(α + λ + µ)2 + (1 − α)2 u22 ]
A + Bu22
=
,
2C
where
A = 2α(α + λ + µ)2 − (1 − α)(α + λ + µ),
B = 2α(1 − α)2 − (1 − α)(α + λ + µ),
C = (α + λ + µ)2 + (1 − α)2 u22 > 0.
We note that Re Ψ(iu2 , v1 ) ≤ 0 if and only if, A ≤ 0 and B ≤ 0. From A ≤ 0,
we obtain α as given by (2.1) and B ≤ 0 gives us 0 ≤ α < 1, and this completes
the proof.
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Khalida Inayat Noor
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Theorem 2.2. For λ > −1, µ > 0 and (λ + µ) > 0,
1, α), where α is given by (2.1).
Vk (λ, µ, 0) ⊂ Vk (λ, µ +
Proof. Let f ∈ Vk (λ, µ, 0). Then Lµλ f ∈ Vk (0) = Vk and, by (1.7) z(Lµλ )0 ∈
Rk (0) = Rk . This implies
Lµλ (zf 0 ) ∈ Rk
=⇒
zf 0 ∈ Rk (λ, µ, 0) ⊂ Rk (λ, µ + 1, α).
Consequently f ∈ Vk (λ, µ + 1, α), where α is given by (2.1).
On Analytic Functions Related
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Theorem 2.3. Let λ > −1, µ > 0 and (λ + µ) > 0. Then
Tk (λ, µ, β, 0) ⊂ Tk (λ, µ + 1, γ, α),
Khalida Inayat Noor
where α is given by (2.1) and γ ≤ β is defined in the proof.
Proof. Let f ∈ Tk (λ, µ, 0). Then there exists g ∈ R2 (λ, µ, 0) such that
∈ Pk (β), for z ∈ E, 0 ≤ β < 1. Let
0
z(Lµ+1
λ f (z))
Lµ+1
λ g(z)
n
0
z(Lµ
λf )
µ
Lλ g
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o
= (1 − γ)p(z) + γ
k 1
k 1
=
+
{(1 − γ)p1 (z) + γ} −
−
{(1 − γ)p2 (z) + γ},
4 2
4 2
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where p(0) = 1, and p(z) is analytic in E.
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Making use of (2.2) and Theorem 2.1 with k = 2, we have
z(Lµλ f (z))0
(2.3)
−β
Lµλ g(z)
(1 − γ)zp0 (z)
= (1 − γ)p(z) + (γ − β) +
∈ Pk ,
(1 − α)q(z) + α + λ + µ
and q ∈ P, where
(1 − α)q(z) + α =
0
z Lµ+1
λ g(z)
Lµ+1
λ g(z)
,
z ∈ E.
On Analytic Functions Related
to Certain Family of Integral
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Khalida Inayat Noor
Using (1.6), we form the functional Φ(u, v) by taking u = u1 +iu2 = pi (z), v =
v1 + iv2 = zp0i in (2.3) as
(2.4)
(1 − γ)v
Φ(u, v) = (1 − γ)u + (γ − β) +
.
(1 − α)q(z) + α + λ + µ
It can be easily seen that the function Φ(u, v) defined by (2.4) satisfies the conditions (i) and (ii) of Lemma 1.1. To verify the condition (iii), we proceed, with
q(z) = q1 + iq2 , as follows:
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Re [Φ(iu2 , v1 )]
(1 − γ)v1
(1 − α)(q1 + iq2 ) + α + λ + µ
(1 − γ)(1 − α)v1 q1 + (1 − γ)(α + λ + µ)v1
= (γ − β) +
[(1 − α)q1 + α + λ + µ]2 + (1 − α)2 q22
= (γ − β) + Re
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≤ (γ − β) −
≤ 0,
for
1 (1 − γ)(1 − α)(1 + u22 )q1 + (1 − γ)(α + λ + µ)(1 + u22 )
2
[(1 − α)q1 + α + λ + µ]2 + (1 − α)2 q22
γ ≤ β < 1.
Therefore, applying Lemma 1.1, pi ∈ P, i = 1, 2 and consequently p ∈ Pk and
thus f ∈ Tk (λ, µ + 1, γ, α).
Using the same technique and relation (1.8) with Theorem 2.3, we have the
following.
Tk? (λ, µ, β, 0)
Theorem 2.4. For λ > −1, µ > 0, λ + µ > 0,
1, γ, α), where γ and α are as given in Theorem 2.3.
⊂
Tk? (λ, µ
+
Remark 1. For different choices of k, λ and µ, we obtain several interesting
special cases of the results proved in this paper.
On Analytic Functions Related
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Khalida Inayat Noor
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References
[1] S.S. MILLER, Differential inequalities and Carathéordary functions, Bull.
Amer. Math. Soc., 81 (1975), 79–81.
[2] K. INAYAT NOOR, On close-to-convex and related functions, Ph.D Thesis,
University of Wales, U.K., 1972.
[3] K. INAYAT NOOR, On quasi-convex functions and related topics, Int. J.
Math. Math. Sci., 10 (1987), 241–258.
[4] K.S. PADMANABHAN AND R. PARVATHAM, Properties of a class of
functions with bounded boundary rotation, Ann. Polon. Math., 31 (1975),
311–323.
[5] B. PINCHUK, Functions with bounded boundary rotation, Israel J. Math.,
10 (1971), 7–16.
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