Research

# Integral operators on new families of meromorphic functions of complex order

Aabed Mohammed and Maslina Darus*

Author Affiliations

School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, 43600 Bangi, Selangor Darul Ehsan, Malaysia

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Journal of Inequalities and Applications 2011, 2011:121  doi:10.1186/1029-242X-2011-121

 Received: 21 August 2011 Accepted: 25 November 2011 Published: 25 November 2011

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### 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.

##### Keywords:
Analytic function; meromorphic function; integral operator

### 1 Introduction

Let Σ denote the class of functions of the form:

(1.1)

which are analytic in the punctured open unit disk

(1.2)

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:

(1.3)

A function f ∈ Σ is a meromorphic convex function of order δ (0 ≤ δ < 1), if f satisfies the following inequality:

(1.4)

and we denote this class by Σk(δ).

For f ∈ Σ, Wang et al. [1] and Nehari and Netanyahu [2] introduced and studied the subclass ΣN(β) of Σ consisting of functions f(z) satisfying

(1.5)

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

(1.6)

where , 0 ≤ δ < 1.

Definition 1.2 Let a function f ∈ Σ be analytic in . Then f is in the class ΣKb(δ) if, and only if, f satisfies

(1.7)

where b C\{0}, 0 ≤ δ < 1. We note that f ∈ Σ Kb(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

(1.8)

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

(1.9)

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

(1.10)

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

(1.11)

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

(1.12)

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

(1.13)

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

(1.14)

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:

(1.15)

and

(1.16)

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

(1.17)

then f ∈ Σ*(δ).

Corollary 1.2 If f ∈ Σ satisfies the following inequality

(1.18)

then f ∈ Σ*.

Corollary 1.3 If f ∈ Σ satisfies the following inequality

(1.19)

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 γi > 0, fi ∈ Σ and

(2.1)

If

(2.2)

then , where β > 1.

Proof On successive differentiation of , which is defined in (1.5), we get

(2.3)

and

(2.4)

Then from (2.3) and (2.4), we obtain

(2.5)

By multiplying (2.5) with z yield

(2.6)

This is equivalent to

(2.7)

Therefore, we have

(2.8)

Then, we easily get

(2.9)

Taking real parts of both sides of (2.9), we obtain

(2.10)

Let

(2.11)

Since , applying Corollary 1.1, we have

(2.12)

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 γi > 0, fi ∈ Σ and

(2.13)

If

(2.14)

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 γi > 0, fi ∈ Σ and

(2.15)

If

(2.16)

then , where β > 1.

Theorem 2.4 For i ∈ {1,..., n}, let γi > 0, fi ∈ Σ and

(2.17)

If

(2.18)

then , where β > 1.

Theorem 2.5 For i ∈ {1,..., n}, let γi > 0 and (0 ≤ δ < 1 and ).

If

(2.19)

then is in the class .

Proof From (2.7), we have

(2.20)

Equivalently, (2.20) can be written as

(2.21)

Taking the real part of both terms of the last expression, we have

(2.22)

Since , hence

(2.23)

so that

(2.24)

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 γi > 0 and (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and ). If

(2.25)

then is in the class .

Proof Since , it follows from Definition 1.3 that

(2.26)

Considering Definition 1.8 and with the help of (2.21), we obtain

(2.27)

This completes the proof.   □

Theorem 2.7 For i ∈ {1,..., n}, let γi > 0 and (α > 0 and ). If

(2.28)

then is in the class .

Proof Since , it follows from Definition 1.5 that

(2.29)

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 γi > 0, fi ∈ Σ and

(3.1)

If , then , β > 1.

Proof On successive differentiation of , which is defined in (1.16), we have

(3.2)

and

(3.3)

Then from (3.2) and (3.3), we obtain

(3.4)

By multiplying (3.4) with z yields,

(3.5)

that is equivalent to

(3.6)

Therefore, we have

(3.7)

so that

(3.8)

Taking the real parts of both terms of the last expression, we obtain

(3.9)

Let

(3.10)

Since , fi ∈ Σk(δ), we get

(3.11)

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 γi > 0, fi ∈ Σ and

(3.12)

If , then , β > 1.

Proof It follows from (3.7) that

(3.13)

Taking the real parts of both terms of the last expression, we obtain

(3.14)

Thus, we have

(3.15)

Let

(3.16)

Since and , we have

(3.17)

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 γi > 0 fi ∈ ΣKb(δi) (0 ≤ δ < 1 and )). If

(3.18)

then is in the class .

Theorem 3.4 For i ∈ {1,..., n}, let γi > 0 and (α ≥ 0, δ ∈ [-1,1), α + δ ≥ 0 and )). If

(3.19)

then is in the class .

Theorem 3.5 For i ∈ {1,..., n}, let γi > 0 and (α ≥ 0, and )). If

(3.20)

then is in the class .

### Competing interests

The authors declare that they have no competing interests.

### Authors' contributions

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.

### Acknowledgements

The above study was supported by MOHE grant: UKM-ST-06-FRGS0244-2010.

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