Journal of Inequalities in Pure and
Applied Mathematics
ON CERTAIN CLASSES OF MEROMORPHIC FUNCTIONS
INVOLVING INTEGRAL OPERATORS
volume 7, issue 4, article 138,
2006.
KHALIDA INAYAT NOOR
Mathematics Department
COMSATS Institute of Information Technology
Islamabad, Pakistan.
Received 06 October, 2006;
accepted 15 November, 2006.
Communicated by: N.E. Cho
EMail: khalidanoor@hotmail.com
Abstract
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c
2000
Victoria University
ISSN (electronic): 1443-5756
252-06
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Abstract
We introduce and study some classes of meromorphic functions defined by using a meromorphic analogue of Noor [also Choi-Saigo-Srivastava] operator for
analytic functions. Several inclusion results and some other interesting properties of these classes are investigated.
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
2000 Mathematics Subject Classification: 30C45, 30C50.
Key words: Meromorphic functions, Functions with positive real part, Convolution,
Integral operator, Functions with bounded boundary and bounded radius
rotation, Quasi-convex and close-to-convex functions.
Khalida Inayat Noor
This research is supported by the Higher Education Commission, Pakistan, through
research grant No: 1-28/HEC/HRD/2005/90.
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
3
8
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1.
Introduction
Let M denote the class of functions of the form
∞
1 X
f (z) = +
an z n ,
z n=0
which are analytic in D = {z : 0 < |z| < 1}.
Let Pk (β) be the class of analytic functions p(z) defined in unit disc E =
D ∪ {0}, satisfying the properties p(0) = 1 and
Z 2π Re p(z) − β (1.1)
1 − β dθ ≤ kπ,
0
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
Khalida Inayat Noor
iθ
where z = re , k ≥ 2 and 0 ≤ β < 1. When β = 0, we obtain the class Pk
defined in [14] and for β = 0, k = 2, we have the class P of functions with
positive real part.
Also, we can write (1.1) as
Z
1 2π 1 + (1 − 2β)ze−it
(1.2)
p(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
(1.3)
dµ(t) = 2, and
|dµ(t)| ≤ k.
0
0
From (1.1), we can write, for p ∈ Pk (β),
k 1
k 1
+
p1 (z) −
−
p2 (z),
(1.4)
p(z) =
4 2
4 2
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where p1 , p2 ∈ P2 (β) = P (β), z ∈ E.
We define the function λ(a, b, z) by
∞
1 X (a)n+1 n
λ(a, b, z) = +
z ,
z n=0 (c)n+1
z ∈ D,
c 6= 0, −1, −2, . . . , a > 0, where (a)n is the Pochhamer symbol (or the shifted
factorial) defined by
(a)0 = 1,
(a)n = a(a + 1) · · · (a + n − 1),
n > 1.
We note that
1
λ(a, c, z) = 2 F1 (1, a; c, z),
z
2 F1 (1, a; c, z) is Gauss hypergeometric function.
Let f ∈ M. Denote by L̃(a, c); M −→ M, the operator defined by
L̃(a, c)f (z) = λ(a, c, z) ? f (z),
z ∈ D,
where the symbol ? stands for the Hadamard product (or convolution). The
operator L̃(a, c) was introduced and studied in [5]. This operator is closely
related to the Carlson-Shaeffer operator [1] defined for the space of analytic
and univalent functions in E, see [11, 13].
We now introduce a function (λ(a, c, z))(−1) given by
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
Khalida Inayat Noor
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λ(a, c, z) ? (λ(a, c, z))(−1) =
1
,
z(1 − z)µ
(µ > 0),
z ∈ D.
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Analogous to L̃(a, c), a linear operator Iµ (a, c) on M is defined as follows, see
[2].
Iµ (a, c)f (z) = (λ(a, c, z))(−1) ? f (z),
(µ > 0, a > 0, c 6= 0, −1, −2, . . . , z ∈ D).
(1.5)
We note that
I2 (2, 1)f (z) = f (z),
and
I2 (1, 1)f (z) = zf 0 (z) + 2f (z).
It can easily be verified that
(1.6)
z (Iµ (a + 1, c)f (z))0 = aIµ (a, c)f (z) − (a + 1)Iµ (a + 1, c)f (z),
(1.7)
z (Iµ (a, c)f (z))0 = µIµ+1 (a, c)f (z) − (µ + 1)Iµ (a, c)f (z).
We note that the operator Iµ (a, c) is motivated essentially by the operators defined and studied in [2, 11].
Now, using the operator Iµ (a, c), we define the following classes of meromorphic functions for µ > 0, 0 ≤ η, β < 1, α ≥ 0, z ∈ D.
We shall assume, unless stated otherwise, that a 6= 0, −1, −2, . . . , c 6=
0, −1, −2, . . .
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
Khalida Inayat Noor
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Definition 1.1. A function f ∈ M is said to belong to the class M Rk (η) for
z ∈ D, 0 ≤ η < 1, k ≥ 2, if and only if
zf 0 (z)
∈ Pk (η)
−
f (z)
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and f ∈ M Vk (η), for z ∈ D, 0 ≤ η < 1, k ≥ 2, if and only if
−
(zf 0 (z))0
∈ Pk (η).
f 0 (z)
We call f ∈ M Rk (η), a meromorphic function with bounded radius rotation of
order η and f ∈ M Vk a meromorphic function with bounded boundary rotation.
Definition 1.2. Let f ∈ M, 0 ≤ η < 1, k ≥ 2, z ∈ D. Then
f ∈ M Rk (µ, η, a, c) if and only if
Iµ (a, c)f ∈ M Rk (η).
Also
f ∈ M Vk (µ, η, a, c) if and only if
Iµ (a, c)f ∈ M Vk (η),
z ∈ D.
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
Khalida Inayat Noor
We note that, for z ∈ D,
f ∈ M Vk (µ, η, a, c)
⇐⇒
−zf 0 ∈ M Rk (µ, η, a, c).
Definition 1.3. Let α ≥ 0, f ∈ M, 0 ≤ η, β < 1, µ > 0 and z ∈ D. Then
f ∈ Bkα (µ, β, η, a, c), if and only if there exists a function g ∈ M C(µ, η, a, c),
such that
(Iµ (a, c)f (z))0
(z(Iµ (a, c)f (z))0 )0
(1 − α)
+α −
∈ Pk (β).
(Iµ (a, c)g(z))0
(Iµ (a, c)g(z))0
In particular, for α = 0, k = a = µ = 2, and c = 1, we obtain the class of
meromorphic close-to-convex functions, see [4]. For α = 1, k = µ = a =
2, c = 1, we have the class of meromorphic quasi-convex functions defined for
z ∈ D. We note that the class C ? of quasi-convex univalent functions, analytic
in E, were first introduced and studied in [7]. See also [9, 12].
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The following lemma will be required in our investigation.
Lemma 1.1 ([6]). Let u = u1 + iu2 and v = v1 + iv2 and let Φ(u, v) be a
complex-valued function satisfying the conditions:
(i) Φ(u, v) is continuous in a domain D ⊂ C 2 ,
(ii) (1, 0) ∈ D and Φ(1, 0) > 0.
2
(iii) Re Φ(iu2 , v1 ) ≤ 0 whenever (iu2 , v1 ) ∈ D and v1 ≤ − 12 (1 + u2 ).
P
m
0
If h(z) = 1+ ∞
m=1 cm z is a function, analytic in E, such that (h(z), zh (z)) ∈
D and Re(h(z), zh0 (z)) > 0 for z ∈ E, then Re h(z) > 0 in E.
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
Khalida Inayat Noor
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2.
Main Results
Theorem 2.1.
M Rk (µ + 1, η, a, c) ⊂ M Rk (µ, β, a, c) ⊂ M Rk (µ, γ, a + 1, c).
Proof. We prove the first part of the result and the second part follows by using
similar arguments. Let
f ∈ M Rk (µ + 1, η, a, c),
z∈D
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
and set
(2.1)
k 1
k 1
H(z) =
+
h1 (z) −
−
h2 (z)
4 2
4 2
z(Iµ (a, c)f (z))0
=−
,
Iµ (a, c)f (z)
Khalida Inayat Noor
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where H(z) is analytic in E with H(0) = 1.
Simple computation together with (2.1) and (1.7) yields
z (Iµ+1 (a, c)f (z))0
(2.2) −
Iµ+1 (a, c)f (z)
zH 0 (z)
= H(z) +
∈ Pk (η),
−H(z) + µ + 1
Let
"
Φµ (z) =
∞
X
#
"
∞
X
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#
1
µ
1
1
+
zk +
+
kz k ,
µ + 1 z k=0
µ + 1 z k=0
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then
zH 0 (z)
(H(z) ? zΦµ (z)) = H(z) +
−H(z) + µ + 1
k 1
k 1
=
+
(h1 (z) ? zΦµ (z)) −
−
(h2 (z) ? zΦµ (z))
4 2
4 2
zh01 (z)
k 1
+
h1 (z) +
=
4 2
−h (z) + µ + 1
1
k 1
zh02 (z)
(2.3)
−
−
h2 (z) +
.
4 2
−h2 (z) + µ + 1
Since f ∈ M Rk (µ + 1, η, a, c), it follows from (2.2) and (2.3) that
zh0i (z)
hi (z) +
∈ P (η), i = 1, 2, z ∈ E.
−hi (z) + µ + 1
Let hi (z) = (1 − β)pi (z) + β. Then
(1 − β)zp0i (z)
(1 − β)pi (z) +
+ (β − η) ∈ P,
−(1 − β)pi (z) − β + µ + 1
Khalida Inayat Noor
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z ∈ E.
We shall show that pi ∈ P, i = 1, 2.
We form the functional Φ(u, v) by taking u = pi (z), v = zp0i (z) with u =
u1 + iu2 , v = v1 + iv2 . The first two conditions of Lemma 1.1 can easily be
verified. We proceed to verify the condition (iii).
Φ(u, v) = (1 − β)u +
(1 − β)v
+ (β − η),
−(1 − β)u − β + µ + 1
On Certain Classes of
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Involving Integral Operators
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implies that
Re Φ(iu2 , v1 ) = (β − η) +
(1 − β)(1 + µ − β)v1
.
(1 + µ − β)2 + (1 − β)2 u22
By taking v1 ≤ − 12 (1 + u22 ), we have
Re Φ(iu2 , v1 ) ≤
A + Bu22
,
2C
where
A = 2(β − η)(1 + µ − β)2 − (1 − β)(1 + µ − β),
B = 2(β − η)(1 − β)2 − (1 − β)(1 + µ − β),
C = (1 + µ − β)2 + (1 − β)2 u22 > 0.
We note that Re Φ(iu2 , v1 ) ≤ 0 if and only if A ≤ 0 and B ≤ 0. From A ≤ 0,
we obtain
i
p
1h
(3 + 2µ + 2η) − (3 + 2µ + 2η)2 − 8 ,
(2.4)
β=
4
and B ≤ 0 gives us 0 ≤ β < 1.
Now using Lemma 1.1, we see that pi ∈ P for z ∈ E, i = 1, 2 and hence
f ∈ M Rk (µ, β, a, c) with β given by (2.4).
In particular, we note that
i
p
1h
β=
(3 + 2µ) − 4µ2 + 12µ + 1 .
4
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
Khalida Inayat Noor
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Theorem 2.2.
M Vk (µ + 1, η, a, c) ⊂ M Vk (µ, β, a, , c) ⊂ M Vk (µ, γ, a + 1, c).
Proof.
f ∈ M Vk (µ + 1, η, a, c)
⇐⇒ −zf 0 ∈ M Rk (µ + 1, η, a, c)
⇒ −zf 0 ∈ M Rk (µ, β, a, c)
⇐⇒ f ∈ M Vk (µ, β, a, c),
where β is given by (2.4).
The second part can be proved with similar arguments.
On Certain Classes of
Meromorphic Functions
Involving Integral Operators
Khalida Inayat Noor
Theorem 2.3.
Bkα (µ + 1, β1 , η1 , a, c) ⊂ Bkα (µ, β2 , η2 , a, c) ⊂ Bkα (µ, β3 , η3 , a + 1, c),
where ηi = ηi (βi , µ), i = 1, 2, 3 are given in the proof.
Proof. We prove the first inclusion of this result and other part follows along
similar lines. Let f ∈ Bkα (µ + 1, β1 , η1 , a, c). Then, by Definition 1.3, there
exists a function g ∈ M V2 (µ + 1, η1 , a, c) such that
(Iµ+1 (a, c)f (z))0
(z(Iµ+1 (a, c)f (z))0 )0
(2.5) (1 − α)
+α −
∈ Pk (β1 ).
(Iµ+1 (a, c)g(z))0
(Iµ+1 (a, c)g(z))0
Set
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(2.6)
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p(z) = (1 − α)
0
(Iµ (a, c)f (z))
(Iµ (a, c)g(z))0
(z(Iµ (a, c)f (z))0 )0
+α −
,
(Iµ (a, c)g(z))0
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where p is an analytic function in E with p(0) = 1.
Now, g ∈ M V2 (µ + 1, η1 , a, c) ⊂ M V2 (µ, η2 , a, c), where η2 is given by the
equation
2η22 + (3 + 2µ − 2η1 )η2 − [2η1 (1 + µ) + 1] = 0.
(2.7)
Therefore,
q(z) =
(z(Iµ (a, c)g(z))0 )0
−
(Iµ (a, c)g(z))0
∈ P (η2 ),
By using (1.7), (2.5), (2.6) and (2.7), we have
zp0 (z)
(2.8)
p(z) + α
∈ Pk (β1 ),
−q(z) + µ + 1
z ∈ E.
On Certain Classes of
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Involving Integral Operators
Khalida Inayat Noor
q ∈ P (η2 ),
z ∈ E.
Contents
With
p(z) =
k 1
+
4 2
[(1 − β2 )p1 (z) + β2 ] −
k 1
−
4 2
[(1 − β2 )p2 (z) + β2 ] ,
(2.8) can be written as
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k 1
+
4 2
(1 − β2 )zp01 (z)
(1 − β2 )p1 (z) + α
+ β2
−q(z) + µ + 1
k 1
(1 − β2 )zp02 (z)
−
(1 − β2 )p2 (z) + α
+ β2 ,
−
4 2
−q(z) + µ + 1
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where
(1 − β2 )zp0i (z)
+ β2 ∈ P (β1 ),
(1 − β2 )pi (z) + α
−q(z) + µ + 1
That is
(1 − β2 )zp0i (z)
(1 − β2 )pi (z) + α
+ (β2 − β1 ) ∈ P,
−q(z) + µ + 1
z ∈ E, i = 1, 2.
z ∈ E,
i = 1, 2.
We form the functional Ψ(u, v) by taking u = u1 + iu2 = pi , v = v1 + iv2 =
zp0i , and
On Certain Classes of
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(1 − β2 )v
Ψ(u, v) = (1 − β2 )u + α
+ (β2 − β1 ),
(−q1 + iq2 ) + µ + 1
(q = q1 + iq2 ).
The first two conditions of Lemma 1.1 are clearly satisfied. We verify (iii), with
v1 ≤ − 12 (1 + u22 ) as follows
Re Ψ(iu2 , v1 )
α(1 − β2 )v1 {(−q1 + µ + 1) + iq2 }
= (β2 − β1 ) + Re
(−q + µ + 1)2 + q22
2(β − 2 − β1 )| − q + µ + 1|2 − α(1 − β2 )(−q1 + µ + 1)(1 + u22 )
≤
2| − q + µ + 1|2
2
A + Bu2
=
, C = | − q + µ + 1|2 > 0
2C
≤ 0, if A ≤ 0 and B ≤ 0,
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where
A = 2(β2 − β1 )| − q + µ + 1|2 − α(1 − β2 )(−q1 + µ + 1),
B = −α(1 − β2 )(−q1 + µ + 1) ≤ 0.
From A ≤ 0, we get
β2 =
(2.9)
2β1 | − q + µ + 1|2 + α Re(−q(z) + µ + 1)
.
2| − q + µ + 1|2 + α Re(−q(z) + µ + 1)
Hence, using Lemma 1.1, it follows that p(z), defined by (2.6), belongs to
Pk (β2 ) and thus f ∈ Bkα (µ, β2 , η2 , a, c), z ∈ D. This completes the proof
of the first part. The second part of this result can be obtained by using similar
arguments and the relation (1.6).
Theorem 2.4.
(i)
(ii)
Bkα (µ, β, η, a, c) ⊂ Bk0 (µ, γ, η, a, c)
Bkα1 (µ, β, η, a, c) ⊂ Bkα2 (µ, β, η, a, c),
On Certain Classes of
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Khalida Inayat Noor
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for
0 ≤ α2 < α1 .
Proof. (i). Let
(Iµ (a, c)f (z))0
h(z) =
,
(Iµ (a, c)g(z))0
h(z) is analytic in E and h(0) = 1. Then
(z(Iµ (a, c)f (z))0 )0
(Iµ (a, c)f (z))0
+α −
(2.10) (1 − α)
(Iµ (a, c)g(z))0
(Iµ (a, c)g(z))0
zh0 (z)
= h(z) + α
,
−h0 (z)
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where
(z(Iµ (a, c)g(z))0 )0
h0 (z) = −
∈ P (η).
(Iµ (a, c)g(z))0
Since f ∈ Bkα (µ, β, η, a, c), it follows that
zh0 (z)
h(z) + α
∈ Pk (β), h0 ∈ P (η),
−h0 (z)
Let
h(z) =
k 1
+
4 2
h1 (z) −
Then (2.10) implies that
zh0i (z)
hi (z) + α
∈ P (β),
−h0 (z)
k 1
−
4 2
for z ∈ E.
h2 (z).
On Certain Classes of
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Involving Integral Operators
Khalida Inayat Noor
z ∈ E,
i = 1, 2,
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and from use of similar arguments, together with Lemma 1.1, it follows that
hi ∈ P (γ), i = 1, 2, where
γ=
2β|h0 |2 + α Re h0
.
2|h0 |2 + α Re h0
Therefore h ∈ Pk (γ), and f ∈ Bk0 (µ, γ, η, a, c), z ∈ D. In particular, it can
be shown that hi ∈ P (β), i = 1, 2. Consequently h ∈ Pk (β) and f ∈
Bk0 (µ, β, η, a, c) in D.
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For α2 = 0, we have (i). Therefore, we let α2 > 0 and f ∈ Bkα1 (µ, β, η, a, c).
There exist two functions H1 , H2 ∈ Pk (β) such that
(Iµ (a, c)f (z))0
(z(Iµ (a, c)f (z))0 )0
H1 (z) = (1 − α1 )
+ α1 −
(Iµ (a, c)g(z))0
(Iµ (a, c)g(z))0
(Iµ (a, c)f (z))0
H2 (z) =
, g ∈ M V2 (µ, η, a, c).
(Iµ (a, c)g(z))0
Now
(z(Iµ (a, c)f (z))0 )0
(Iµ (a, c)f (z))0
(2.11) (1 − α2 )
+ α2 −
(Iµ (a, c)g(z))0
(Iµ (a, c)g(z))0
α2
α2
= H1 (z) + 1 −
H2 (z).
α1
α1
Since the class Pk (β) is a convex set [10], it follows that the right hand side of
(2.11) belongs to Pk (β) and this shows that f ∈ Bkα2 (µ, β, η, a, c) for z ∈ D.
This completes the proof.
Let f ∈ M, b > 0 and let the integral operator Fb be defined by
Z z
b
(2.12)
Fb (f ) = Fb (f )(z) = b+1
tb f (t)dt.
z
0
From (2.12), we note that
(2.13)
z (Iµ (a, c)Fb (f )(z))0 = bIµ (a, c)f (z) − (b + 1)Iµ (a, c)Fb (f )(z).
Using (2.12), (2.13) with similar techniques used earlier, we can prove the following:
On Certain Classes of
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Involving Integral Operators
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Theorem 2.5. Let f ∈ M Rk (µ, β, a, c), or M Vk (µ, β, a, c), or Bkα (µ, β, η, a, c),
for z ∈ D. Then Fb (f ) defined by (2.12) is also in the same class for z ∈ D.
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References
[1] B.C. CARLSON AND D.B. SCHAEFFER, Starlike and prestarlike hypergeometric functions, SIAM J. Math. Anal., 15 (1984), 737–745.
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Involving Integral Operators
[4] V. KUMAR AND S.L. SHULKA, Certain integrals for classes of p-valent
meromorphic functions, Bull. Austral. Math. Soc., 25 (1982), 85–97.
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[5] J.L. LIU AND H.M. SRIVASTAVA, A linear operator and associated families of meromorphically multivalued functions, J. Math. Anal. Appl., 259
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[6] S.S. MILLER, Differential inequalities and Caratheodory functions, Bull.
Amer. Math. Soc., 81 (1975), 79–81.
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[7] K.I. NOOR, On close-to-conex and related functions, Ph.D Thesis, University of Wales, Swansea, U. K., 1972.
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[8] K.I. NOOR, A subclass of close-to-convex functions of order β type γ,
Tamkang J. Math., 18 (1987), 17–33.
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[9] K.I. NOOR, On quasi-convex functions and related topics, Inter. J. Math.
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[10] K.I. NOOR, On subclasses of close-to-convex functions of higher order,
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[12] K.I. NOOR AND D.K. THOMAS, On quasi-convex univalent functions,
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