Journal of Inequalities in Pure and
Applied Mathematics
GENERALIZED INTEGRAL OPERATOR AND MULTIVALENT
FUNCTIONS
volume 6, issue 2, article 51,
2005.
KHALIDA INAYAT NOOR
Mathematics Department
COMSATS Institute of Information Technology
Islamabad, Pakistan.
Received 20 February, 2005;
accepted 02 March, 2005.
Communicated by: Th.M. Rassias
EMail: khalidanoor@hotmail.com
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
061-05
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Abstract
P
p+j analytic in the
Let A(p) be the class of functions f : f(z) = z p + ∞
j=1 aj z
zp
open unit disc E. Let, for any integer n > −p, fn+p−1 (z) = (1−z)
n+p . We define
(−1)
(−1)
P
z
fn+p−1 (z) by using convolution ? as fn+p−1 (z) ? fn+p−1 (z) = (1−z)
n+p . A func
R 2π Rep(z)−ρ tion p, analytic in E with p(0) = 1, is in the class Pk (ρ) if 0 p−ρ dθ ≤ kπ,
where z = reiθ , k ≥ 2 and 0 ≤ ρ < p. We use the class Pk (ρ) to introduce
a new class of multivalent analytic functions and define an integral operator
(−1)
In+p−1 (f) = fn+p−1 ? f(z) for f(z) belonging to this class. We derive some
interesting properties of this generalized integral operator which include inclusion results and radius problems.
2000 Mathematics Subject Classification: Primary 30C45, 30C50.
Key words: Convolution (Hadamard product), Integral operator, Functions with positive real part, Convex functions.
Contents
1
2
3
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1
Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
Generalized Integral Operator
and Multivalent Functions
Khalida Inayat Noor
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1.
Introduction
Let A(p) denote the class of functions f given by
p
f (z) = z +
∞
X
aj z p+j ,
p ∈ N = {1, 2, . . .}
j=1
which are analytic in the unit disk E = {z : |z| < 1}. The Hadamard product
or convolution (f ? g) of two functions with
f (z) = z p +
∞
X
aj,1 z p+j
and g(z) = z p +
j=1
∞
X
z p+j
(f ? g)(z) = z p +
∞
X
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aj,1 aj,2 z p+j .
Contents
j=1
The integral operator In+p−1 : A(p) −→ A(p) is defined as follows, see [2].
(−1)
zp
For any integer n greater than −p, let fn+p−1 (z) = (1−z)
n+p and let fn+p−1 (z)
be defined such that
(−1)
fn+p−1 (z) ? fn+p−1 (z) =
zp
.
(1 − z)p+1
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Then
(1.2)
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j=1
is given by
(1.1)
Generalized Integral Operator
and Multivalent Functions
In+p−1 f (z) =
(−1)
fn+p−1 (z)
zp
? f (z) =
(1 − z)n+p
(−1)
? f (z).
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From (1.1) and (1.2) and a well known identity for the Ruscheweyh derivative
[1, 8], it follows that
(1.3)
z (In+p f (z))0 = (n + p)In+p−1 f (z) − nIn+p f (z).
For p = 1, the identity (1.3) is given by Noor and Noor [3].
Let Pk (ρ) be the class of functions p(z) analytic in E satisfying the properties p(0) = 1 and
Z 2π Re p(z) − ρ (1.4)
p − ρ dθ ≤ kπ,
0
where z = reiθ , k ≥ 2 and 0 ≤ ρ < p. For p = 1, this class was introduced in
[5] and for ρ = 0, see [6]. For ρ = 0, k = 2, we have the well known class P
of functions with positive real part and the class k = 2 gives us the class P (ρ)
of functions with positive real part greater than ρ. Also from (1.4), we note that
p ∈ Pk (ρ) if and only if there exist p1 , p2 ∈ Pk (ρ) such that
k 1
k 1
(1.5)
p(z) =
+
p1 (z) −
−
p2 (z).
4 2
4 2
It is known [4] that the class Pk (ρ) is a convex set.
Definition 1.1. Let f ∈ A(p). Then f ∈ Tk (α, p, n, ρ) if and only if
In+p−1 f (z)
In+p f (z)
(1 − α)
+α
∈ Pk (ρ),
zp
zp
for α ≥ 0, n > −p, 0 ≤ ρ < p, k ≥ 2 and z ∈ E.
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2.
Preliminary Results
Lemma 2.1. Let p(z) = 1 + b1 z + b2 z 2 + · · · ∈ P (ρ). Then
Re p(z) ≥ 2ρ − 1 +
2(1 − ρ)
.
1 + |z|
This result is well known.
Lemma 2.2 ([7]). If p(z) is analytic in E with p(0) = 1 and if λ1 is a complex number satisfying Re λ1 ≥ 0, (λ1 =
6 0), then Re{p(z) + λ1 zp0 (z)} >
β (0 ≤ β < p) implies
Generalized Integral Operator
and Multivalent Functions
Khalida Inayat Noor
Re p(z) > β + (1 − β)(2γ1 − 1),
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where γ1 is given by
Z
γ1 =
1
Re λ1 −1
1+t
Contents
dt.
0
Lemma 2.3 ([9]). If p(z) is analytic in E, p(0) = 1 and Re p(z) > 21 , z ∈
E, then for any function F analytic in E, the function p ? F takes values in the
convex hull of the image E under F.
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3.
Main Results
Theorem 3.1. Let f ∈ Tk (α, p, n, ρ1 ) and g ∈ Tk (α, p, n, ρ2 ), and let F = f ?g.
Then F ∈ Tk (α, p, n, ρ3 ) where


n+p
)−1
Z 1 ( 1−α
u
n+p
(3.1)
ρ3 = 1 − 4(1 − ρ1 )(1 − ρ2 ) 1 −
du .
1−α 0 1+u
This results is sharp.
Proof. Since f ∈ Tk (α, p, n, ρ1 ), it follows that
In+p−1 f (z)
In+p f (z)
H(z) = (1 − α)
+α
∈ Pk (ρ1 ),
zp
zp
and so using (1.3), we have
(3.2)
In+p f (z) =
n+p (
z
1−α
)
z
Z
t
n+p
1−α −1
H(t)dt.
0
n+p
)
n + p −( 1−α
In+p g(z) =
z
1−α
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n+p
)
n + p −( 1−α
In+p F (z) =
z
1−α
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z
Z
t
n+p
1−α −1
H ? (t)dt,
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where H ? ∈ Pk (ρ2 ).
Using (3.1) and (3.2), we have
(3.4)
Khalida Inayat Noor
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n+p
− 1−α
Similarly
(3.3)
Generalized Integral Operator
and Multivalent Functions
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Z
z
t
0
n+p
1−α −1
Q(t)dt,
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where
(3.5)
k 1
k 1
Q(z) =
+
q1 (z) −
−
q2 (z)
4 2
4 2
n+p Z z n+p
)
n + p −( 1−α
1−α −1
z
t
(H ? H ? )(t)dt.
=
1−α
0
Now
k 1
k 1
H(z) =
+
h1 (z) −
−
h2 (z)
4 2
4 2
k 1
k 1
?
?
H(z) =
+
h1 (z) −
−
h?2 (z),
4 2
4 2
(3.6)
where hi ∈ P (ρ1 ) and h?i ∈ Pk (ρ2 ), i = 1, 2.
Since
1
h?i (z) − ρ2 1
?
pi (z) =
+ ∈P
,
2(1 − ρ2 )
2
2
(3.7)
Khalida Inayat Noor
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i = 1, 2,
we obtain that (hi ? p?i )(z) ∈ P (ρ1 ), by using the Herglotz formula.
Thus
(hi ? h?i )(z) ∈ P (ρ3 )
with
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ρ3 = 1 − 2(1 − ρ1 )(1 − ρ2 ).
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Using (3.4), (3.5), (3.6), (3.7) and Lemma 2.1, we have
Z
n+p
−1
n + p 1 1−α
Re qi (z) =
u
Re{(hi ? h?i )(uz)}du
1−α 0
Z
n+p
−1
n + p 1 1−α
2(1 − ρ3 )
≥
u
2ρ3 − 1 +
du
1−α 0
1 + u|z|
Z
n+p
−1
2(1 − ρ3 )
n + p 1 1−α
du
u
2ρ3 − 1 +
>
1+u
1−α 0


n+p
Z 1 1−α
−1
n+p
u
= 1 − 4(1 − ρ1 )(1 − ρ2 ) 1 −
du .
1−α 0 1+u
From this we conclude that F ∈ Tk (α, p, n, ρ3 ), where ρ3 is given by (3.1).
We discuss the sharpness as follows:
We take
k 1 1 − (1 − 2ρ1 )z
k 1 1 + (1 − 2ρ1 )z
H(z) =
+
−
−
,
4 2
1−z
4 2
1+z
k 1 1 + (1 − 2ρ2 )z
k 1 1 − (1 − 2ρ2 )z
?
H (z) =
+
−
−
.
4 2
1−z
4 2
1+z
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Since
1 + (1 − 2ρ2 )z
1 + (1 − 2ρ1 )z
?
1−z
1−z
= 1 − 4(1 − ρ1 )(1 − ρ2 ) +
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4(1 − ρ1 )(1 − ρ2 )
,
1−z
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it follows from (3.5) that
Z
n+p
−1
n + p 1 1−α
4(1 − ρ1 )(1 − ρ2 )
qi (z) =
1 − 4(1 − ρ1 )(1 − ρ2 ) +
u
du
1−α 0
1 − uz


n+p
−1
Z 1 1−α


n+p
u
−→ 1 − 4(1 − ρ1 )(1 − ρ2 ) 1 −
du as z −→ 1.

1−α 0 1+u 
This completes the proof.
We define Jc : A(p) −→ A(p) as follows:
Z
c + p z c−1
(3.8)
Jc (f ) =
t f (t)dt,
zc 0
Generalized Integral Operator
and Multivalent Functions
Khalida Inayat Noor
where c is real and c > −p.
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Theorem 3.2. Let f ∈ Tk (α, p, n, ρ) and Jc (f ) be given by (3.8). If
In+p f (z)
In+p Jc (f )
(3.9)
(1 − α)
+α
∈ Pk (ρ),
zp
zp
Contents
then
In+p Jc (f )
zp
∈ Pk (γ),
z∈E
and
(3.10)
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γ = ρ(1 − ρ)(2σ − 1)
−1
Z 1
Re 1−α
λ+p
σ=
1+t
dt.
0
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Proof. From (3.8), we have
(c + p)In+p f (z) = cIn+p Jc (f ) + z(In+p Jc (f ))0 .
Let
(3.11)
Hc (z) =
k 1
+
4 2
s1 (z) −
k 1
−
4 2
s2 (z) =
In+p Jc (f )
.
zp
From (3.9), (3.10) and (3.11), we have
In+p f (z)
In+p Jc (f )
1−α 0
(1 − α)
+α
= Hc (z) +
zH (z)
zp
zp
λ+p c
Khalida Inayat Noor
and consequently
1−α 0
si (z) +
zs (z) ∈ P (ρ),
λ+p i
i = 1, 2.
Using Lemma 2.2, we have Re{si (z)} > γ where γ is given by (3.10). Thus
Hc (z) =
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Let
n+p
Jn (f (z)) := Jn (f ) =
zp
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In+p Jc (f )
∈ Pk (γ)
zp
and this completes the proof.
(3.12)
Generalized Integral Operator
and Multivalent Functions
Z
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z
n−1
t
f (t)dt.
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Then
In+p−1 Jn (f ) = In+p (f ),
and we have the following.
J. Ineq. Pure and Appl. Math. 6(2) Art. 51, 2005
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Theorem 3.3. Let f ∈ Tk (α, p, n+1, ρ). Then Jn (f ) ∈ Tk (α, p, n, ρ) for z ∈ E.
Theorem 3.4. Let φ ∈ Cp , where Cp is the class of p-valent convex functions,
and let f ∈ Tk (α, p, n, ρ). Then φ ? f ∈ Tk (α, p, n, ρ) for z ∈ E.
Proof. Let G = φ ? f. Then
In+p−1 G(z)
In+p G(z)
+α
p
z
zp
In+p (φ ? f )(z)
In+p−1 (φ ? f )(z)
= (1 − α)
+α
p
z
zp
In+p−1 f (z)
In+p f (z)
φ(z)
= p ? (1 − α)
+α
z
zp
zp
φ(z)
= p ? H(z), H ∈ Pk (ρ)
z
k 1
φ(z)
=
+
(p − ρ)
? h1 (z) + ρ
4 2
zp
φ(z)
k 1
−
−
(p − ρ)
? h2 (z) + ρ , h1 , h2 ∈ P.
4 2
zp
n
o
Since φ ∈ Cp , Re φ(z)
> 12 , z ∈ E and so using Lemma 2.3, we conclude
zP
that G ∈ Tk (α, p, n, ρ).
(1 − α)
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3.1.
Applications
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(1) We can write Jc (f ) defined by (3.8) as
J. Ineq. Pure and Appl. Math. 6(2) Art. 51, 2005
Jc (f ) = φc ? f,
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where φc is given by
∞
X
p+c m
φc (z) =
z ,
m+c
m=p
(c > −p)
and φc ∈ Cp . Therefore, from Theorem 3.4, it follows that Jc (f ) ∈
Tk (α, p, n, ρ).
(2) Let Jn (f ), defined by (3.12), belong to Tk (α, p, n, ρ). Then f ∈ Tk (α, p, n, ρ)
√
for |z| < rn = 2+(1+n)
. In fact, Jn (f ) = Ψn ? f, where
3+n2
Ψn (z) = z p +
∞
X
n+j−1
j=2
n+1
Khalida Inayat Noor
z j+p−1
Title Page
n
zp
1
zp
=
·
+
·
n + 1 1 − z n + 1 (1 − z)2
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and Ψn ∈ Cp for
|z| < rn =
1+n
√
.
2 + 3 + n2
Tk (α1 , p, n, ρ) ⊂ Tk (α2 , p, n, ρ),
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Now In+p−1 Jn (f ) = Ψn ? In+p−1 f, and using Theorem 3.4, we obtain the
result.
Theorem 3.5. For 0 ≤ α2 < α1 ,
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Proof. For α2 = 0, the proof is immediate. Let α2 > 0 and let f ∈ Tk (α1 , p, n, ρ).
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Then
In+p−1 f (z)
In+p f (z)
(1 − α2 )
+
α
2
p
zp
z α2
α1
In+p−1 f (z)
In+p−1 f (z)
In+p−1 f (z)
+
−1
+ (1 − α1 )
+ α1
α1
α2
zp
zp
zp
α2
α2
= 1−
H1 (z) + H2 (z), H1 , H2 ∈ Pk (ρ).
α1
α1
Since Pk (ρ) is a convex set, we conclude that f ∈ Tk (α2 , p, n, ρ) for z ∈ E.
Theorem 3.6. Let f ∈ Tk (0, p, n, ρ). Then f ∈ Tk (α, p, n, ρ) for
|z| < rα =
2α +
√
1
,
4α2 − 2α + 1
1
α 6= ,
2
0 < α < 1.
Generalized Integral Operator
and Multivalent Functions
Khalida Inayat Noor
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Proof. Let
p
p
z
z
+α
1−z
(1 − z)2
∞
X
= zp +
(1 + (m − 1)α)z m+p−1 .
Ψα (z) = (1 − α)
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Ψα ∈ Cp for
|z| < rα =
Page 13 of 15
2α +
√
1
4α2 − 2α + 1
1
α 6= ,
2
0<α<1
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We can write
In+p−1 f (z)
In+p f (z)
Ψα (z) In+p−1 f (z)
(1 − α)
+α
=
?
.
p
p
z
z
zp
zp
Applying Theorem 3.4, we see that f ∈ Tk (α, p, n, ρ) for |z| < rα .
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References
[1] R.M. GOEL AND N.S. SOHI, A new criterion for p-valent functions, Proc.
Amer. Math. Soc., 78 (1980), 353–357.
[2] J.L. LIU AND K. INAYAT NOOR, Some properties of Noor integral operator, J. Natural Geometry, 21 (2002), 81–90.
[3] K. INAYAT NOOR AND M.A. NOOR, On integral operators, J. Math. Anal.
Appl., 238 (1999), 341–352.
[4] K. INAYAT NOOR, On subclasses of close-to-convex functions of higher
order, Internat. J. Math. Math. Sci., 15(1992), 279–290.
[5] K.S. PADMANABHAN AND R. PARVATHAM, Properties of a class of
functions with bounded boundary rotation, Ann. Polon. Math., 31 (1975),
311–323.
[6] B. PINCHUK, Functions with bounded boundary rotation, Isr. J. Math., 10
(1971), 7–16.
[7] S. PONNUSAMY, Differential subordination and Bazilevic functions,
Preprint.
[8] S. RUSCHEWEYH, New criteria for univalent functions, Proc. Amer. Math.
Soc., 49 (1975), 109–115.
[9] R. SINGH AND S. SINGH, Convolution properties of a class of starlike
functions, Proc. Amer. Math. Soc., 106 (1989), 145–152.
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