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Abstract and Applied Analysis
Volume 2011, Article ID 159405, 10 pages
http://dx.doi.org/10.1155/2011/159405
Research Article

A New Class of Meromorphically Analytic Functions with Applications to the Generalized Hypergeometric Functions

1Faculty of Management, Multimedia University, Cyberjaya, 63100 Selangor, Malaysia
2School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi, 43600 Selangor, Malaysia

Received 27 March 2011; Accepted 7 July 2011

Academic Editor: Yoshikazu Giga

Copyright © 2011 Firas Ghanim and Maslina Darus. 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 a new subclass of meromorphically analytic functions, which is defined by means of a Hadamard product (or convolution). A characterization property such as the coefficient bound is obtained for this class. The other related properties, which are investigated in this paper, include the distortion and the radius of starlikeness. We also consider several applications of our main results to the generalized hypergeometric functions.

1. Introduction

Let be the class of functions which are analytic in the open unit disk As usual, we denote by the subclass of , consisting of functions which are also univalent in .

Let be a fixed point in and . In [1], Kanas and Ronning introduced the following classes Later, Acu and Owa [2] studied the classes extensively.

The class is defined by geometric property that the image of any circular arc centered at is starlike with respect to , and the corresponding class is defined by the property that the image of any circular arc centered at is convex. We observed that the definitions are somewhat similar to the ones introduced by Goodman in [3, 4] for uniformly starlike and convex functions except that, in this case, the point is fixed.

Let denote the subclass of consisting of the function of the form The functions in are said to be starlike functions of order if and only if for some . We denote by the class of all starlike functions of order . Similarly, a function in is said to be convex of order if and only if for some . We denote by the class of all convex functions of order .

For the function , we define and, for , we can write where and .

The differential operator is studied extensively by Ghanim and Darus [5, 6] and Ghanim et al. [7].

The Hadamard product or convolution of the functions given by (1.3) with the function and given, respectively, by can be expressed as follows: Suppose that and are two analytic functions in the unit disk . Then, we say that the function is subordinate to the function , and we write if there exists a Schwarz function with and such that By applying the above subordination definition, we introduce here a new class of meromorphically functions, which is defined as follows:

Definition 1.1. A function of the form (1.3) is said to be in the class if it satisfies the following subordination property: where , with condition .

The purpose of this paper is to investigate the coefficient estimates, distortion properties, and the radius of starlikeness for the class . Some applications of the main results involving generalized hypergeometric functions are also considered.

2. Characterization and Other Related Properties

In this section, we begin by proving a characterization property which provides a necessary and sufficient condition for a function of the form (1.3) to belong to the class of meromorphically analytic functions.

Theorem 2.1. The function is said to be a member of the class if it satisfies The equality is attained for the function given by

Proof. Let , and suppose that Then, in view of (2.2), we have Letting , we get which is equivalent to our condition of the theorem, so that . Hence we have the theorem.

Theorem 2.1 immediately yields the following result.

Corollary 2.2. If the function belongs to the class , then , where the equality holds true for the functions given by (2.2).

We now state the following growth and distortion properties for the class .

Theorem 2.3. If the function defined by (1.3) is in the class , then for , one has

Proof. Since , Theorem 2.1 readily yields the inequality Thus, for and utilizing (2.8), we have Also, from Theorem 2.1, we get Hence This completes the proof of Theorem 2.3.

We next determine the radius of meromorphically starlikeness of the class , which is given by Theorem 2.4.

Theorem 2.4. If the function defined by (1.3) is in the class , then is meromorphically starlike of order in the disk , where The equality is attained for the function given by (2.2).

Proof. It suffices to prove that For , we have Hence (2.14) holds true for or With the aid of (2.1) and (2.16), it is true to say that for fixed Solving (2.17) for , we obtain This completes the proof of Theorem 2.4.

3. Applications Involving Generalized Hypergeometric Functions

Let us define the function by for , and , where is the Pochhammer symbol. We note that where Corresponding to the function and using the Hadamard product which was defined earlier in the introduction section for , we define here a new linear operator on by For a function , we define and, for , We note studied by Ghanim and Darus [5, 6] and Ghanim et al. [7], and also, studied by Ghanim and Darus [8, 9] and Ghanim et al. [10].

The subordination relation (1.12) in conjunction with (3.4) and (3.6) takes the following form: .

Definition 3.1. A function of the form (1.3) is said to be in the class if it satisfies the subordination relation (3.7) above.

Theorem 3.2. The function is said to be a member of the class if it satisfies The equality is attained for the function given by .

Proof. By using the same technique employed in the proof of Theorem 2.1 along with Definition 3.1, we can prove Theorem 3.2.

The following consequences of Theorem 3.2 can be deduced by applying (3.8) and (3.9) along with Definition 3.1.

Corollary 3.3. If the function belongs to the class , then , where the equality holds true for the functions given by (3.9).

Corollary 3.4. If the function defined by (1.3) is in the class , then is meromorphically starlike of order in the disk , where The equality is attained for the function given by (3.9).

A slight background related to the formation of the present operator can be found in [11], and other work can be tackled using this type of operator. Also, the meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava [12, 13], Liu [14], Liu and Srivastava [15], and Cho and Kim [16].

Acknowledgment

The work presented here was fully supported by UKM-ST-06-FRGS0244-2010.

References

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