Research Article | Open Access

Ibtisam Aldawish, Maslina Darus, "Some Properties for an Integral Operator Defined by Generalized Hypergeometric Function", *Journal of Applied Mathematics*, vol. 2014, Article ID 134910, 8 pages, 2014. https://doi.org/10.1155/2014/134910

# Some Properties for an Integral Operator Defined by Generalized Hypergeometric Function

**Academic Editor:**Song Cen

#### Abstract

We define a new general integral operator for meromorphic functions involving the generalized hypergeometric function. Furthermore, we study the characterization and other properties for this operator.

#### 1. Introduction

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

We say that a function is meromorphic starlike of order and belongs to the class , if it satisfies the following inequality: A function is a meromorphic convex function of order , if satisfies the following inequality: and we denote this class by .

Also, let be the subclass of , consisting of the functions , which satisfy the inequality where the subclass was introduced and studied by Wang et al. [1] and Nehari and Netanyahu [2].

A -hypergeometric function is a power series in one complex variable with power series coefficients which depend (apart from ) on complex upper parameters , , as follows: with , where when , , and is the Pochhammer symbol defined by Tracing back the history of basic hypergeometric series defined by (5) has brought us to Heine dated in 1846. Therefore it is sometimes called Heine’s series. For brief survey on -hypergeometric functions, one may refer to [3–5] (see also [6–8]).

For , , and , the basic hypergeometric function defined in (5) takes the form which converges absolutely in the open unit disk .

Corresponding to the function , let Analogous to the differential operator defined in [9] which involves the -hypergeometric functions on the normalized analytic functions, we define the following differential operator on the space of meromorphic functions in the class by If , then from (9) we may deduce that where , and .

It should be remarked that the operator given by (10) is a generalization of many other operators considered earlier; for example, consider the following.(i)For , , , , the operator was defined by Frasin and Darus [10] and studied by El-Ashwah and Aouf [11].(ii)For , , , , , , the operator was imposed by Liu and Srivastava [12].(iii)For , , , , , the operator was introduced and studied by Liu and Srivastava [13]. Further, we note in passing that this operator is closely related to the Carlson-Shaffer operator defined on the space of analytic univalent functions in .(iv)For , , , , , the operator , where is the differential operator, was introduced by Ganigi and Uralegaddi [14], and then it was generalized by Yang [15].Now, making use of the differential operator given by (10), we introduce the following integral operator in the class .

*Definition 1. *Let , . We define the integral operator by

*Remark 2. *Note that for , , , , we obtain the integral operator defined by Mohammed and Darus [16]; see also [17–19].

With the aid of differential operator given by (10), we define a new subclass of functions in as follows.

*Definition 3. *Let a function be analytic in . Then is in the class if, and only if, satisfies
where is defined in (10) and .

*Definition 4. *Let a function be analytic in . Then is in the class if, and only if, satisfies
where is defined in (10) and , .

*Definition 5. *Let a function be analytic in . Then is in the class if, and only if, satisfies
where is defined in (10) and .

*Definition 6. *Let a function be analytic in . Then is in the class if, and only if, satisfies
where is defined in (10) and .

*Remark 7. *Putting , , and in Definition 3, we obtain the result corresponding to the class mentioned in (3).

For , , and in Definitions 4, 5, and 6, we obtain and the classes of meromorphic functions, introduced and studied by Mohammed and Darus [20].

#### 2. Preliminary Definitions

We begin by recalling each of the following definitions of subclasses of meromorphic functions , and which will be required in our investigation.

*Definition 8 (see [20]). *Let a function be analytic in . Then is in the class if, and only if, satisfies
where , .

*Definition 9 (see [20]). *Let a function be analytic in . Then is in the class if, and only if, satisfies
where .

*Definition 10 (see [20]). *Let a function be analytic in . Then is in the class if, and only if, satisfies
where .

#### 3. Main Results

In this section, we study some properties for the integral operator defined by (12) of the subclasses given by Definitions 3, 4, 5, and 6.

Theorem 11. *Let , for and . If
**
and , then is in the class , .*

*Proof. *By differentiating the equality given by (12), we get
Again, differentiating (21), we deduce that
By using (21) and (23), we have
Multiplying (24) by yields that
or, equivalently, that
Thus, we get
That is,
Taking the real parts of both terms of the last expression and using the fact that for all , we impose
Setting
and since
we conclude that
Hence, by (20), we have , so that , which evidently completes the proof of Theorem 11.

Theorem 12. *Let , for , and . If
**
then is in the class , where .*

*Proof. *We know from the proof of Theorem 11 that
which is equivalent to
Taking the real part of both terms of (35), we get
Now, since , we have
Letting
and in view of (33), it follows that , so that . This completes the proof of Theorem 12.

Theorem 13. *Let , for , and . If
**
then is in the class .*

*Proof. *We want to show that
We find from (26) and (36) that
Since , it follows from Definition 5 that
From hypothesis (39), we note that
Hence . Therefore, we complete the proof of Theorem 13.

Theorem 14. *Let for , and . If
**
then is in the class .*

*Proof. *Our aim is to prove that
With the aid of (26), (36) can be written as follows:
Since , it follows from Definition 6 and by a simple calculation that
which means that , and we obtain the proof of the theorem.

#### Conflict of Interests

The authors declare that they have no competing interests.

#### Authors’ Contribution

Ibtisam Aldawish and Maslina Darus read and approved the final paper.

#### Acknowledgment

The work presented here was partially supported by UKM’s Grant FRGSTOPDOWN/2013/ST06/UKM/01/1.

#### References

- Z.-G. Wang, Y. Sun, and Z.-H. Zhang, “Certain classes of meromorphic multivalent functions,”
*Computers & Mathematics with Applications*, vol. 58, no. 7, pp. 1408–1417, 2009. View at: Publisher Site | Google Scholar | MathSciNet - Z. Nehari and E. Netanyahu, “On the coefficients of meromorphic schlicht functions,”
*Proceedings of the American Mathematical Society*, vol. 8, pp. 15–23, 1957. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - G. Gasper and M. Rahman,
*Basic Hypergeometric Series*, vol. 35 of*Encyclopedia of Mathematics and Its Applications*, Cambridge University Press, Cambridge, UK, 1990. View at: MathSciNet - H. Exton,
*$q$-Hypergeometric Functions and Applications*, Ellis Horwood, Chichester, UK, 1983. View at: MathSciNet - H. A. Ghany, “q-derivative of basic hypergeometric series with respect to parameters,”
*International Journal of Mathematical Analysis*, vol. 3, no. 33, pp. 1617–1632, 2009. View at: Google Scholar | MathSciNet - R. W. Ibrahim, “On certain linear operator defined by basic hypergeometric functions,”
*Matematichki Vesnik*, vol. 65, no. 1, pp. 1–7, 2013. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. W. Ibrahim and M. Darus, “On analytic functions associated with the Dziok-SRIvastava linear operator and Srivastava-Owa fractional integral operator,”
*Arabian Journal for Science and Engineering*, vol. 36, no. 3, pp. 441–450, 2011. View at: Publisher Site | Google Scholar | MathSciNet - I. Aldawish and M. Darus, “New subclass of analytic function associated with the generalized hypergeometric functions,”
*Electronic Journal of Mathematical Analysis and Applications*, vol. 2, no. 2, pp. 163–171, 2014. View at: Google Scholar - H. Aldweby and M. Darus, “Properties of a subclass of analytic functions defined by a generalized operator involving
*q*− Hypergeometric function,”*Far East Journal of Mathematical Sciences*, vol. 81, no. 2, pp. 189–200, 2013. View at: Google Scholar - B. A. Frasin and M. Darus, “On certain meromorphic functions with positive coefficients,”
*Southeast Asian Bulletin of Mathematics*, vol. 28, no. 4, pp. 615–623, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet - R. M. El-Ashwah and M. K. Aouf, “Hadamard product of certain meromorphic starlike and convex functions,”
*Computers & Mathematics with Applications*, vol. 57, no. 7, pp. 1102–1106, 2009. View at: Publisher Site | Google Scholar | MathSciNet - J.-L. Liu and H. M. Srivastava, “Classes of meromorphically multivalent functions associated with the generalized hypergeometric function,”
*Mathematical and Computer Modelling*, vol. 39, no. 1, pp. 21–34, 2004. View at: Publisher Site | Google Scholar | MathSciNet - J.-L. Liu and H. M. Srivastava, “A linear operator and associated families of meromorphically multivalent functions,”
*Journal of Mathematical Analysis and Applications*, vol. 259, no. 2, pp. 566–581, 2001. View at: Publisher Site | Google Scholar | MathSciNet - M. R. Ganigi and B. A. Uralegaddi, “New criteria for meromorphic univalent functions,”
*Bulletin Mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie. Nouvelle Série*, vol. 33, no. 81, pp. 9–13, 1989. View at: Google Scholar | MathSciNet - D. Yang, “On a class of meromorphic starlike multivalent functions,”
*Bulletin of the Institute of Mathematics Academia Sinica*, vol. 24, no. 2, pp. 151–157, 1996. View at: Google Scholar | MathSciNet - A. Mohammed and M. Darus, “A new integral operator for meromorphic functions,”
*Acta Universitatis Apulensis*, no. 24, pp. 231–238, 2010. View at: Google Scholar | MathSciNet - A. Mohammed and M. Darus, “Starlikeness properties of a new integral operator for meromorphic functions,”
*Journal of Applied Mathematics*, vol. 2011, Article ID 804150, 8 pages, 2011. View at: Publisher Site | Google Scholar | MathSciNet - A. Mohammed and M. Darus, “Some properties of certain integral operators on new subclasses of analytic functions with complex order,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 161436, 9 pages, 2012. View at: Publisher Site | Google Scholar | MathSciNet - A. Mohammed and M. Darus, “The order of starlikeness of new $p$-valent meromorphic functions,”
*International Journal of Mathematical Analysis*, vol. 6, no. 27, pp. 1329–1340, 2012. View at: Google Scholar | MathSciNet - A. Mohammed and M. Darus, “Integral operators on new families of meromorphic functions of complex order,”
*Journal of Inequalities and Applications*, vol. 2011, article 121, 12 pages, 2011. View at: Publisher Site | Google Scholar | MathSciNet

#### Copyright

Copyright © 2014 Ibtisam Aldawish 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.