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Integral operators on new families of meromorphic functions of complex order
Journal of Inequalities and Applications volume 2011, Article number: 121 (2011)
Abstract
In this article, we define and investigate new families of certain subclasses of meromorphic functions of complex order. Considering the new subclasses, several properties for certain integral operators are derived.
2010 Mathematics Subject Classification: 30C45.
1 Introduction
Let Σ denote the class of functions of the form:
which are analytic in the punctured open unit disk
where is the open unit disk .
We say that a function f ∈ Σ is meromorphic starlike of order δ (0 ≤ δ < 1), and belongs to the class Σ*(δ), if it satisfies the inequality:
A function f ∈ Σ is a meromorphic convex function of order δ (0 ≤ δ < 1), if f satisfies the following inequality:
and we denote this class by Σ k (δ).
For f ∈ Σ, Wang et al. [1] and Nehari and Netanyahu [2] introduced and studied the subclass Σ N (β) of Σ consisting of functions f(z) satisfying
Let denote the class of functions f of the form , which are analytic in the open unit disk .
Analogous to several subclasses [3–10] of analytic functions of , we define the following subclasses of Σ.
Definition 1.1 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where , 0 ≤ δ < 1.
Definition 1.2 Let a function f ∈ Σ be analytic in . Then f is in the class ΣK b (δ) if, and only if, f satisfies
where b ∈ C\{0}, 0 ≤ δ < 1. We note that f ∈ Σ K b (δ) if, and only if, .
Furthermore, the classes
are the classes of meromorphic starlike functions of order δ and meromorphic convex functions of order δ in , respectively. Moreover, the classes
are familiar classes of starlike and convex functions in , respectively.
Definition 1.3 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, .
Definition 1.4 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, .
We note that if, and only if, .
For α = 0, we have
Definition 1.5 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α > 0, .
Definition 1.6 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α > 0, .
We note that if, and only if, .
Let us also introduce the following families of new subclasses , , and as follows.
Definition 1.7 Let a function f ∈ Σ be analytic in . Then f is in the class , if, and only if, f satisfies
where , 0 ≤ δ < 1.
We note that if, and only if, .
Definition 1.8 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0, .
We note that if, and only if, .
Definition 1.9 Let a function f ∈ Σ be analytic in . Then f is in the class if, and only if, f satisfies
where α > 0, .
We note that if, and only if, .
Recently, many authors introduced and studied various integral operators of analytic and univalent functions in the open unit disk [11–21].
Most recently, Mohammed and Darus [22, 23] introduced the following two general integral operators of meromorphic functions Σ:
and
Goyal and Prajapat [24] obtained the following results for f ∈ Σ to be in the class Σ*(δ), 0 ≤ δ < 1.
Corollary 1.1 If f ∈ Σ satisfies the following inequality
then f ∈ Σ*(δ).
Corollary 1.2 If f ∈ Σ satisfies the following inequality
then f ∈ Σ*.
Corollary 1.3 If f ∈ Σ satisfies the following inequality
then f ∈ Σ*.
In this article, we derive several properties for the integral operators and of the subclasses given by (1.5) and Definitions 1.1 to 1.6.
2 Some properties for
In this section, we investigate some properties for the integral operator defined by (1.15) of the subclasses given by (1.5), Definitions 1.1, 1.3, and 1.5.
Theorem 2.1 For i ∈ {1,..., n}, let γ i > 0, f i ∈ Σ and
If
then , where β > 1.
Proof On successive differentiation of , which is defined in (1.5), we get
and
Then from (2.3) and (2.4), we obtain
By multiplying (2.5) with z yield
This is equivalent to
Therefore, we have
Then, we easily get
Taking real parts of both sides of (2.9), we obtain
Let
Since , applying Corollary 1.1, we have
Then, by the hypothesis (2.2), we have β > 1. Therefore, , where β > 1. This completes the proof. □
Letting δ = 0 in Theorem 2.1, we have
Corollary 2.2 For i ∈ {1,..., n}, let γ i > 0, f i ∈ Σ and
If
then , where β > 1.
Making use of (2.11), Corollary 1.2 and Corollary 1.3, one can prove the following results.
Theorem 2.3 For i ∈ {1,..., n}, let γ i > 0, f i ∈ Σ and
If
then , where β > 1.
Theorem 2.4 For i ∈ {1,..., n}, let γ i > 0, f i ∈ Σ and
If
then , where β > 1.
Theorem 2.5 For i ∈ {1,..., n}, let γ i > 0 and (0 ≤ δ < 1 and ).
If
then is in the class .
Proof From (2.7), we have
Equivalently, (2.20) can be written as
Taking the real part of both terms of the last expression, we have
Since , hence
so that
Then .
Now, adopting the techniques used by Breaz et al. [11] and Bulut [21], we prove the following two theorems.
Theorem 2.6 For i ∈ {1,..., n}, let γ i > 0 and (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and ). If
then is in the class .
Proof Since , it follows from Definition 1.3 that
Considering Definition 1.8 and with the help of (2.21), we obtain
This completes the proof. □
Theorem 2.7 For i ∈ {1,..., n}, let γ i > 0 and (α > 0 and ). If
then is in the class .
Proof Since , it follows from Definition 1.5 that
Considering this inequality and (2.21), we obtain
This completes the proof. □
3 Some properties for
In this section, we investigate some properties for the integral operator defined by (1.16) of subclasses given by (1.5), Definitions 1.2, 1.4, and 1.6.
Theorem 3.1 For i ∈ {1,..., n}, let γ i > 0, f i ∈ Σ and
If , then , β > 1.
Proof On successive differentiation of , which is defined in (1.16), we have
and
Then from (3.2) and (3.3), we obtain
By multiplying (3.4) with z yields,
that is equivalent to
Therefore, we have
so that
Taking the real parts of both terms of the last expression, we obtain
Let
Since , f i ∈ Σ k (δ), we get
which, in light of the hypothesis (3.1), yields β > 1.
Therefore, , β > 1. This completes the proof. □
Theorem 3.2 For i ∈ {1,..., n}, let γ i > 0, f i ∈ Σ and
If , then , β > 1.
Proof It follows from (3.7) that
Taking the real parts of both terms of the last expression, we obtain
Thus, we have
Let
Since and , we have
Then, by the hypothesis (3.12), we see that β > 1. Therefore, , β > 1. This completes the proof. □
Now, using the method given in the proofs of Theorems 2.5, 2.6, and 2.7, one can prove the following results:
Theorem 3.3 For i ∈ {1,..., n}, let γ i > 0 f i ∈ ΣK b (δ i ) (0 ≤ δ < 1 and )). If
then is in the class .
Theorem 3.4 For i ∈ {1,..., n}, let γ i > 0 and (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and )). If
then is in the class .
Theorem 3.5 For i ∈ {1,..., n}, let γ i > 0 and (α ≥ 0, and )). If
then is in the class .
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Acknowledgements
The above study was supported by MOHE grant: UKM-ST-06-FRGS0244-2010.
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AM is currently a PhD student under supervision of MD and jointly worked on deriving the results. All the authors read and approved the final manuscript.
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Mohammed, A., Darus, M. Integral operators on new families of meromorphic functions of complex order. J Inequal Appl 2011, 121 (2011). https://doi.org/10.1186/1029-242X-2011-121
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DOI: https://doi.org/10.1186/1029-242X-2011-121