Int. J. Open Problems Complex Analysis, Vol. 5, No. 3, November, 2013
c
ISSN 2074-2827; Copyright ICSRS
Publication, 2013
www.i-csrs.org
Majorization for Certain Classes of Meromorphic
Functions Defined by Integral Operator
Kirti Dhuria and Rachana Mathur
Department of Mathematics,
Govt. Dungar (P.G.) College, Bikaner, India
e-mail: [email protected]
e-mail: [email protected]
Abstract
Here we investigate a majorization problem involving starlike
meromorphic function of complex order belonging to a certain subclass of meromorphic univalent function defined by an integral operator introduced recently by Lashin.
Keywords: Meromorphic univalent functions, majorization property, starlike functions, integral operators.
2000 Mathematical Subject Classification: Primary 30C45; Secondary
30C80.
1
Introduction and Preliminaries
Let f (z) and g(z) are analytic in the open unit disk
4 = {z ∈ C and |z| < 1}.
(1.1)
For analytic functions f (z) and g(z) in ∆, we say that f (z) is majorized
by g(z) in ∆ (see [10]) and write
f (z) << g(z) (z ∈ ∆),
(1.2)
if there exists a function φ(z), analytic in ∆ such that |φ(z)| ≤ 1, and
f (z) = φ(z)g(z) (z ∈ ∆).
(1.3)
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Majorization for Certain Classes of
Let Σ denote the class of meromorphic functions of the form
∞
1 X
f (z) = +
ak z k ,
z k=1
(1.4)
which are analytic and univalent in the punctured unit disk
∆∗ := {z ∈ C : 0 < |z| < 1} := ∆ \ {0}
(1.5)
with a simple pole at the origin.
For functions fj ∈ Σ given by
∞
1 X
ak,j z k
fj (z) = +
z k=1
(j = 1, 2; z ∈ ∆∗ ),
(1.6)
we define the Hadamard product (or convolution) of f1 and f2 by
∞
1 X
(f1 ∗ f2 )(z) = +
ak,1 ak,2 z k = (f2 ∗ f1 )(z).
z k=1
(1.7)
Analogous to the operators defined by Jung, Kim and Srivastava [8] on the
normalized analytic functions, Lashin [9] introduced the following integral operators
Qαβ : Σ −→ Σ
defined by
Γ(β + α) 1
Qαβ = Qαβ f (z) =
Γ(β)Γ(α) z β+1
Zz
β
t
t
1−
z
α−1
f (t)dt (α, β > 0; z ∈ ∆∗ ),
0
(1.8)
where Γ(α) is familiar Gamma function.
Using the integral representation of the Gamma function and (1.4), it can
be easily shown that
∞
Qαβ f (z)
1 Γ(α + β) X Γ(k + β + 1)
= +
ak z k
z
Γ(α) k=1 Γ(k + α + β + 1)
(α > 0, β > 0; z ∈ ∆∗).
(1.9)
Obviously
Q1β f (z) := Jβ .
The operator
Jβ : Σ −→ Σ
(1.10)
52
Rachana Mathur and Kirti Dhuria
has also been studied by Lashin [9].
It is easy to verify that (see [9]),
z(Qαβ f (z))0 = (α + β − 1)Qα−1
f (z) − (β + α)Qαβ f (z).
β
(1.11)
Definition 1.3. A function f (z) ∈ Σ is said to be in the class Sβα,j (γ) of
meromorphic functions of complex order γ 6= 0 in ∆ if and only if
(j+1)
z(Qα
β f (z))
1
+j+1
> 0,
< 1− γ
(j)
(Qα
(1.12)
β f (z))
(z ∈ ∆, j ∈ N0 = N ∪ {0}, α > 0, β > 0, γ ∈ C\{0}).
Clearly, we have the following relationships:
(i) Sβ0,0 (γ) = S(γ) (γ ∈ C \ {0}),
(ii) Sβ0,0 (1 − η) = S ∗ (η) (0 ≤ η < 1)
The classes S(γ) and S ∗ (η) are said to be classes of meromorphic starlike
univalent functions of complex order γ 6= 0 and meromorphic starlike univalent
functions of order η (η ∈ < such that 0 ≤ η < 1) in ∆∗ .
An majorization problem for the normalized classes of starlike has been
investigated by Altinas et al. [1] and MacGregor [10]. In the recent paper
of Goyal and Goswami [3] generalized these results for the class of multivalent functions using fractional derivatives operators. Further, Goyal et al. [4],
Goswami and Wang [5], Goswami and Aouf [6], Goswami et al. [7] studied
majorization property for different - different classes. In this paper, we will
study majorization properties for the class of meromorphic functions using integral operator Qαβ .
2. Majorization problems for the class Sβα,j (γ)
Theorem 2.1 Let the function f ∈ Σ and suppose that g ∈ Sβα,j (γ). If
(Qαβ f (z))(j) is majorized by (Qαβ g(z))(j) in ∆∗ , then
|(Qα−1
f (z))(j) | ≤ |(Qα−1
g(z))(j) | f or |z| ≤ r1 (α, β, γ),
β
β
where
r1 (α, β, γ) =
k1 −
(2.1)
p
k 1 2 − 4(β + α − 1)|β + α − 1 + 2γ|
2|β + α − 1 + 2γ|
and
k1 = (β + α + 1 + |β + α − 1 + 2γ|, (β > 0, j ∈ N0 , γ ∈ C \ {0}). (2.2)
Proof. Since g ∈ Sβα,j (γ), we find from (2.1), if
!
(j+1)
1 z(Qαβ g(z))
h1 (z) = 1 −
+ j + 1 (α, β > 0, γ ∈ C \ {0}, j ∈ N0 ),
γ
(Qαβ g(z))(j)
(2.3)
53
Majorization for Certain Classes of
then <{h1 (z)} > 0 (z ∈ ∆) and
h1 (z) =
1 + w(z)
1 − w(z)
(w ∈ Q),
(2.4)
where w(z) = c1 z + c2 z 2 + ... and Q denotes the well known class of bounded
analytic functions in ∆ and satisfies the conditions
w(0) = 0 and |w(z)| ≤ |z| (z ∈ ∆).
Making use of (2.3) and (2.4), we get
z(Qαβ g(z))(j+1)
(Qαβ g(z))(j)
=
(1 + j − 2γ)w(z) − (1 + j)
.
1 − w(z)
(2.5)
By principle of mathematical induction, and (1.11), we easily get
z(Qαβ g(z))(j+1) = (α + β − 1)(Qα−1
g(z))(j) − (α + β + j)(Qαβ g(z))(j) ,
β
(2.6)
(α > 1, β > 0; z ∈ ∆∗ ).
Now using (2.6) in (2.5), we find that
(j)
(α + β − 1)(Qβα−1 g(z))
(Qαβ g(z))(j)
=
= (β + α + j) +
(1 + j − 2γ)w(z) − (1 + j)
1 − w(z)
(α + β − 1) − (α + β − 1 + 2γ)w(z)
1 − w(z)
or
(Qαβ g(z))(j) =
(α + β − 1)(1 − w(z))
(Qα−1 g(z))(j) .
(α + β − 1) − (α + β − 1 + 2γ)w(z) β
(2.7)
Since |w(z)| ≤ |z| (z ∈ ∆), therefore (2.6) yields
α
(j) (Qβ g(z)) ≤
(α + β − 1)[1 + |z|]
(j) α−1
(Qβ g(z)) .
α + β − 1 − |α + β − 1 + 2γ||z|
(2.8)
Next since (Qαβ f (z))(j) is majorized by (Qαβ g(z))(j) in the unit disk ∆∗ , therefore
from (1.3), we have
(Qαβ f (z))(j) = φ(z)(Qαβ g(z))(j)
Differentiating it with respect to ‘z’ and multiplying by ‘z’, we get
z(Qαβ f (z))(j+1) = zϕ0 (z)(Qαβ g(z))(j) + zϕ(z)(Qαβ g(z))(j+1)
54
Rachana Mathur and Kirti Dhuria
Using (2.6), in the above equation, it yields
f (z))(j) =
(Qα−1
β
zϕ0 (z)
(Qα g(z))(j) + ϕ(z)(Qα−1
g(z))(j)
β
(α + β − 1) β
(2.9)
Thus, nothing that ϕ ∈ Q satisfies the inequality (see, e.g. Nehari [6])
|ϕ0 (z)| ≤
1 − |ϕ(z)|2
1 − |z|2
(2.10)
and making use of (2.8) and (2.10) in (2.9), we get
|z|
1 − |ϕ(z)|2
(j) (j) α−1
α−1
(Qβ g(z)) ,
(Qβ f (z)) ≤ |ϕ(z)| +
1 − |z| [α + β − 1 − |2γ + β + α − 1||z|]
(2.11)
which upon setting
|z| = r and |ϕ(z)| =ρ (0 ≤ ρ ≤ 1),
leads us to the inequality
Θ(ρ)
(j) (j) α−1
(Q
g(z))
f
(z))
≤
,
(Qα−1
β
β
(1 − r)(β + α − 1 − |2γ + β + α − 1|r)
where
Θ(ρ) = −rρ2 + (1 − r)(β + α − 1 − |2γ + β + α − 1|r)ρ + r
(2.12)
takes its maximum value at ρ = 1, with r2 = r2 (α, β, γ) where r2 (α, β, γ) is
given by equation (2.2). Furthermore, if 0 ≤ ρ ≤ r2 (α, β, γ), then the function
θ(ρ) defined by
θ(ρ) = −σρ2 + (1 − σ)(β + α − 1 − |2γ + β + α − 1|σ)ρ + σ
(2.13)
is seen to be increasing function on the interval 0 ≤ ρ ≤ 1, so that
θ(ρ) ≤ θ(1) = (1 − σ)(β + α − 1 − |2γ + β + α − 1|σ),
(0 ≤ ρ ≤ 1; 0 ≤ σ ≤ r1 (α, β, γ)).
(2.14)
Hence upon setting ρ = 1, in (2.14), we conclude that (2.1) of Theorem 2.1
holds true for |z| ≤ r1 (α, β, γ), where r1 (α, β, γ) is given by (2.2). This completes the Theorem 2.1.
Setting α = 1 , in Theorem 2.1, we get
Corollary 2.1. Let the function f ∈ Σ and suppose that g ∈ Sβ1,j (γ). If
(Jβ f (z))(j) is majorized by (Jβ g(z))(j) in ∆∗ , then
|(f (z))(j) | ≤ |(g(z))(j) | f or |z| ≤ r2 (α, β, γ),
(2.15)
55
Majorization for Certain Classes of
where
r2 (β, γ) =
k2 −
p
k22 − 4β|β + 2γ|
2|β + 2γ|
and
(k2 = (β + 2 + |β + 2γ|), β > 0, j ∈ N0 , γ ∈ C\{0}).
Further putting β = 1 and γ = 1 − η, j = 0 in Corollary 2.1, we get
Corollary 2.2. Let the function f ∈ Σ and suppose that g ∈ S11,0 (1 − η). If
(J1 f (z)) is majorized by (J1 g(z)) in ∆∗ , then
|f (z)| ≤ |g(z)| f or |z| ≤ r3 ,
(2.16)
where
p
η 2 − 4η + 6
.
r3 =
3−η
For η = 0, the above corollary reduces to the following result :
Corollary 2.3. Let the function f (z) ∈ Σ and suppose that g ∈ S11,0 (1) :=
S11,0 . If (J1 f (z)) is majorized by (J1 g(z)) in ∆∗ , then
√
3− 6
|f (z)| ≤ |g(z)| f or |z| ≤
.
(2.17)
3
3−η−
2
Open Problem
In this paper we studied majorization for the certain class of meromorphic
analytic functions. If we define a class f ∈ Σp such that
f (z) = z
−p
+
∞
X
an+p z n+p , (z ∈ ∆∗ ),
0
then we need to modify integral operator Qαβ for the class of meromorphic
multivalent functions and further using this modified operator we have to find
majorization conditions for modified integral operator.
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Rachana Mathur and Kirti Dhuria
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