ISRN Mathematical Analysis

Volumeย 2012, Article IDย 698307, 11 pages

http://dx.doi.org/10.5402/2012/698307

## On Certain Subclasses of Meromorphic Functions with Positive and Fixed Second Coefficients Involving the Liu-Srivastava Linear Operator

^{1}Research and P.G. Studies in Mathematics, Governt Arts College (Men), Tamilnadu Krishnagiri 635001, India^{2}P.G. Studies in Mathematics, Government Science College, Karnataka Chitradurga 577501, India^{3}Department of Mathematics, Adhiyamaan College of Engineering (Autonomous), Tamilnadu Hosur 635109, India

Received 23 December 2011; Accepted 16 February 2012

Academic Editor: S.ย Zhang

Copyright ยฉ 2012 N. Magesh et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduce and study a subclass of meromorphic univalent functions defined by certain linear operator involving the generalized hypergeometric function. We obtain coefficient estimates, extreme points, growth and distortion inequalities, radii of meromorphic starlikeness, and convexity for the class by fixing the second coefficient. Further, it is shown that the class is closed under convex linear combination.

#### 1. Introduction

Let denote the class of functions of the form which are analytic in the punctured unit disk

Let ,โโ and ,โโ denote the subclasses of that are meromorphically univalent, meromorphically starlike functions of order , and meromorphically convex functions of order , respectively. Analytically, if and only if, is of the form (1.1) and satisfies similarly, , if and only if, is of the form (1.1) and satisfies and similar other classes of meromorphically univalent functions have been extensively studied by Aouf and Darwish [1], Aouf and Joshi [2], Mogra et al. [3], Uralegaddi and Ganigi [4], Uralegaddi and Somanatha[5], and Owa and Pascu [6].

Let , , where is given by (1.1) and is defined by

Then the Hadamard product (or convolution) of the functions and is defined by Let be the class of functions of the form that are analytic and univalent in .

For complex parameters and the generalized hypergeometric function is defined by where denotes the set of all positive integers and is the Pochhammer symbol defined by

Corresponding to a function defined by

Liu and Srivastava [7] considered a linear operator defined by the following Hadamard product (or convolution): where, , , , , ;โโ, . For notational simplicity, we use a shorter notations for and unless otherwise stated in the sequel. We note that the linear operator was earlier defined for multivalent functions by Dziok and Srivastava [8] and was investigated by Liu and Srivastava [7].

Making use of the operator , now we consider a subclass of functions in as follows.

*Definition 1.1. *For and and , let denote a subclass of consisting functions of the form (1.1) satisfying the condition that
where is given by (1.11). Furthermore, we say that a function , whenever is of the form (1.7).

We observe that, by specializing the parameters ,โโ,โโ,โโ,โโ,โโ and the class leads to various subclasses. As for illustrations, we present some examples for the cases.

*Example 1.2. *If and with , , andโโโโof the form (1.7), then we obtain the new subclass โโdefined by

*Example 1.3. *For , , , , and of the form (1.7), then we get the new subclass defined by
where , is the differential operator which was introduced by Ganigi and Uralegaddi [9]. Also we note that the class was introduced by Atshan and Kulkarni [10].

*Example 1.4. *For , ,โโ, and of the form (1.7), then we obtain the new subclass defined by
where the operator was introduced and studied by Liu and Srivastava [11] (see also [12, 13]).

For the class , the following characterization was given by Magesh et al. [14].

Theorem 1.5. *Let be given by (1.7). Then , if and only if,
**
where is given by (1.12).*

For a function defined by (1.7) and in the class , Theorem 1.5, immediately yields Hence we may take

Motivated by Aouf and Darwish [1], Aouf and Joshi [2], Ghanim and Darus [12], and Uralegaddi [15], we now introduce the following class of functions and use the similar techniques to prove our results.

Let be the subclass of consisting functions in of the form

In this paper, coefficient estimates, extreme points, growth and distortion bounds, radii of meromorphically starlikeness and convexity are discussed for the class by fixing the second coefficient. Further, it is shown that the class is closed under convex linear combination.

#### 2. Main Results

In our first theorem, we now find out the coefficient inequality for the class .

Theorem 2.1. *Let the function defined by (1.20). Then , if and only if,
**
The result is sharp.*

*Proof. *By putting
in (1.17), the result is easily derived. The result is sharp for the function

Corollary 2.2. *If the function defined by (1.20) is in the class , then
**
The result is sharp for the function given by (2.3).*

Next we prove the following growth and distortion properties for the class .

Theorem 2.3. *If the function defined by (1.20) is in the class for , then one has
**
The result is sharp for the function given by
*

*Proof. *Since , Theorem 2.1 yields
Thus, for ,
Thus the proof of the theorem is complete.

Theorem 2.4. *If the function defined by (1.20) is in the class for , then one has
**
The result is sharp for the function given by
*

*Proof. *In view of Theorem 2.1, it follows that
Thus, for and making use of (2.11), we obtain

Hence the result follows.

Next, we will show that the class is closed under convex linear combination.

Theorem 2.5. *If
**
and for **
Then if and only if it can expressed in the form
**
where and .*

*Proof. *From (2.13), (2.14), and (2.15), we have
Since
it follows from Theorem 1.5 that the function . Conversely, suppose that . Since
Setting
it follows that
This completes the proof of the theorem.

Theorem 2.6. *The class is closed under linear combination.*

*Proof. *Suppose that the function be given by (1.20), and let the function be given by
Assuming that and are in the class , it is enough to prove that the function defined by
is also in the class . Since
we observe that
with the aid of Theorem 2.1. Thus, .

Next we determine the radii of meromorphically starlikeness and convexity of order for functions in the class .

Theorem 2.7. *Let the function defined by (1.20) be in the class , then *(i)* is meromorphically starlike of order in the disk where , is the largest value for which
*(ii)* is meromorphically convex of order in the disk where , is the largest value for which**
Each of these results is sharp for function given by (2.3).*

*Proof. *It is enough to highlight that
Thus, we have
where denotes , denotes , and denotes .

Hence (2.28) holds true if
or,
and it follows that, from (2.1), we may take
where and

For each fixed , we choose the positive integer for which
is maximal. Then it follows that
Then is starlike of order in provided that
We find the value and the corresponding integer so that
It is the value for which the function is starlike in . (ii) In a similar manner, we can prove our result providing the radius of meromorphic convexity of order for functions in the class , so we skip the details of the proof of (ii).

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