The Scientific World Journal

Volume 2013 (2013), Article ID 164287, 7 pages

http://dx.doi.org/10.1155/2013/164287

## On Harmonic Meromorphic Functions Associated with Basic Hypergeometric Functions

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

Received 4 August 2013; Accepted 14 September 2013

Academic Editors: A. Favini and N. Kallur

Copyright © 2013 Huda Al dweby 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

By making use of basic hypergeometric functions, a class of complex harmonic meromorphic functions with positive coefficients is introduced. We obtain some properties such as coefficient inequality, growth theorems, and extreme points.

#### 1. Introduction and Preliminaries

Recently, the popularity of the study of basic hypergeometric series (also called -hypergeometric series) is increasing among the researchers. Back in 1748, Euler considered the infinite product and ever since it became very important to many areas. However, it was stated in the literature that the development of these functions was slower until Heine (1878) converted a simple observation such that , which then returns the theory of basic hypergeometric series to the famous theory of Gauss hypergeometric series. Various authors [1–7] introduced classes of analytic functions involving hypergeometric functions and investigated their properties. However, all these results concern mostly the Gaussian and generalized hypergeometric functions. It seems that no attempt has been made to derive similar results for the basic hypergeometric functions. In this work, we proceed to define a class of harmonic meromorphic functions in the unit disk associated with the basic hypergeometric function and discuss some of its properties.

The subject of harmonic univalent functions is a recent area of research which was initially established by Clunie and Sheil-Small [8]. The importance of these functions is due to their use in the study of minimal surfaces as well as in various problems related to applied mathematics and perhaps to other areas of sciences. Hengartner and Schober [9] have introduced and studied special classes of harmonic functions, which are defined on the exterior of the unit disk . They have proved that these functions are complex valued harmonic, sense preserving, univalent mappings , admitting the representation where and are analytic in .

For , let denote the class of functions which are harmonic in the punctured unit disk , where and are analytic in and , respectively. The class was studied in [10–12].

We further denote by the subclass of consisting of functions of the form which are univalent harmonic in the punctured unit disk .

A function is said to be in the class of meromorphically harmonic starlike functions in if it satisfies the condition this class has been studied by Jahangiri and Silverman (see [10]). For harmonic functions the convolution is defined by For complex parameters , , , the basic hypergeometric function is defined as follows: with , where when , , denotes the set of positive integers, and is the -shifted factorial defined by By using the ratio test, one recognizes that if , the series (7) converges absolutely (see [13]) for all if and for if .

The -derivative of a function is defined by For a function , observe that then , where is the ordinary derivative. For more properties of , see [14, 15].

Now for , , and , the basic hypergeometric function defined in (7) takes the form which converges absolutely in the open unit disk .

Let where, for convenience, Corresponding to the function , defined in (12), consider and its inverse function defined by The series expansion of the inverse function is given as where Now, let which is analytic function in the punctured unit disk .

Again, corresponding to the function , defined in (12), consider and its inverse function defined by The series expansion of the inverse function is given as Let which is analytic function in . Now, we define the harmonic meromorphic function as where and are of the forms (18) and (22).

Using convolution of harmonic meromorphic functions given by (23) and given by (2), a linear operator is defined as Involving the operator , we introduce the class as follows.

*Definition 1. *Let denote the family of harmonic meromorphic functions satisfying
Equivalently,
for some and for all .

For , , , and , is the class of harmonic meromorphic starlike functions of order and for , , , , and , the class gives the class of harmonic meromorphic starlike functions for all .

Denote .

#### 2. Main Results

Theorem 2. *If is of the form (2) and satisfies the condition
**
where , then is harmonic univalent sense preserving in and . *

*Proof. *Suppose that (27) holds true for . Consider the expression
That is,
The inequality in (28) holds true for all . Therefore, letting in (28), we obtain
By hypothesis (27), it follows that (26) holds, so that . Note that is sense preserving in . This is because
Then, we have . Hence the theorem.

Now, we prove that condition (27) is necessary for functions in .

Theorem 3. *Let be a function defined by (3). Then, if and only if the inequality
**
is satisfied.*

*Proof. *In view of Theorem 2, it suffices to show that the “only if" part is true. Assume that . Then,
Since for all , it follows from (33) that
We now choose the values of on the real axis. Upon the clearing the denominator in (34) and letting through real values, we obtain the following:
which immediately yields the required condition (32).

Next, we consider a distortion property for functions in the class as follows.

Theorem 4. *If defined by (3) is in the class , then for , where , , and , one has
*

*Proof. *We will only prove the right-hand inequality. The proof for the left-hand inequality is similar and we will omit it:
The functions
for show that the bounds given in Theorem 4 are sharp in .

Next, we give the following.

Theorem 5. *Set
**
Then, , if and only if, it can be expressed in the form
**
where , , and . In particular, the extreme points of are and .*

*Proof. *Let
Then,
So, . Conversely, suppose that . Set
Then, by Theorem 3, and .
and note that, by Theorem 3, . Consequently, we obtain
This proves the theorem.

*Remark 6. *Other work related to -hypergeometric functions and analytic functions can be found in [16, 17].

#### Acknowledgment

The work presented here was partially supported by LRGS/TD/2011/UKM/ICT/03/02 and UKM-DLP-2011-050.

#### References

- J. Dziok and H. M. Srivastava, “Classes of analytic functions associated with the generalized hypergeometric function,”
*Applied Mathematics and Computation*, vol. 103, no. 1, pp. 1–13, 1999. View at Google Scholar · View at Scopus - J. A. Kim and K. H. Shon, “Mapping properties for convolutions involving hypergeometric functions,”
*International Journal of Mathematics and Mathematical Sciences*, vol. 2003, no. 17, pp. 1083–1091, 2003. View at Publisher · View at Google Scholar · View at Scopus - K. I. Noor, “Integral operators defined by convolution with hypergeometric functions,”
*Applied Mathematics and Computation*, vol. 182, no. 2, pp. 1872–1881, 2006. View at Publisher · View at Google Scholar · View at Scopus - M. Darus and R. W. Ibrahim, “On integral operator involving generalized hypergeometric function,”
*Advanced Studies in Theoretical Physics*, vol. 4, no. 17-20, pp. 805–816, 2010. View at Google Scholar · View at Scopus - B. C. Carlson and D. B. Shaffer, “Starlike and prestarlike hypergeometric functions,”
*SIAM Journal on Mathematical Analysis*, vol. 15, no. 4, pp. 737–745, 1984. View at Publisher · View at Google Scholar - A. Gangadharan, T. N. Shanmugam, and H. M. Srivastava, “Generalized hypergeometric functions associated with K-uniformly convex functions,”
*Computers and Mathematics with Applications*, vol. 44, no. 12, pp. 1515–1526, 2002. View at Publisher · View at Google Scholar · View at Scopus - Y. C. Kim and H. M. Srivastava, “Fractional integral and other linear operators associated with the Gaussian hypergeometric function,”
*Complex Variables. Theory and Application*, vol. 34, no. 3, pp. 293–312, 1997. View at Publisher · View at Google Scholar - J. Clunie and T. Sheil-Small, “Harmonic univalent functions,”
*Annales Academiae Scientiarum Fennicae. Series A I. Mathematica*, vol. 9, pp. 3–25, 1984. View at Google Scholar - W. Hengartner and G. Schober, “Univalent harmonic functions,”
*Transactions of the American Mathematical Society*, vol. 299, no. 1, pp. 1–31, 1987. View at Publisher · View at Google Scholar - J. M. Jahangiri and H. Silverman, “Meromorphic univalent harmonic functions with negative coefficients,”
*Bulletin of the Korean Mathematical Society*, vol. 36, no. 4, pp. 763–770, 1999. View at Google Scholar - H. Bostanci and M. Öztürk, “New classes of Salagean type meromorphic harmonic functions,”
*International Journal of Mathematics Sciences*, vol. 2, no. 1, pp. 52–57, 2008. View at Google Scholar - K. Al-Shaqsi and M. Darus, “On meromorphic harmonic functions with respect to
*k*-symmetric points,”*Journal of Inequalities and Applications*, vol. 2008, Article ID 259205, 11 pages, 2008. View at Publisher · View at Google Scholar · View at Scopus - G. Gasper and M. Rahman,
*Basic Hypergeometric Series*, vol. 35 of*Encyclopedia of Mathematics and Its Applications*, Cambridge University Press, Cambridge, UK, 1990, With a foreword by Richard Askey. - H. Exton,
*q-hypergeometric functions and applications*, Ellis Horwood Series: Mathematics and Its Applications, Ellis Horwood, Chichester, UK, 1983. - H. A. Ghany, “
*q*-derivative of basic hypergeometric series with respect to parameters,”*International Journal of Mathematical Analysis*, vol. 3, no. 33–36, pp. 1617–1632, 2009. View at Google Scholar - A. Mohammed and M. Darus, “A generalized operator involving the
*q*-hypergeometric function,”*Matematicki Vesnik*, vol. 65, no. 4, pp. 454–465, 2013. View at Google Scholar - H. Aldweby and M. Darus, “Univalence of a new general integral operator associated with the
*q*-hypergeometric function,”*International Journal of Mathematics and Mathematical Sciences*, vol. 2013, Article ID 769537, 5 pages, 2013. View at Google Scholar