International Journal of Mathematics and Mathematical Sciences

Volume 2011, Article ID 459063, 11 pages

http://dx.doi.org/10.1155/2011/459063

## On Certain Class of Analytic Functions Related to Cho-Kwon-Srivastava Operator

^{1}Faculty of Management, Multimedia University, Selangor D. Ehsan, 63100 Cyberjaya, Malaysia^{2}School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Selangor D. Ehsan, 43600 Bangi, Malaysia

Received 27 March 2011; Accepted 29 August 2011

Academic Editor: Stanisława R. Kanas

Copyright © 2011 F. Ghanim and M. 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

Motivated by a multiplier transformation and some subclasses of meromorphic functions which were defined by means of the Hadamard product of the Cho-Kwon-Srivastava operator, we define here a similar transformation by means of the Ghanim and Darus operator. A class related to this transformation will be introduced and the properties will be discussed.

#### 1. Introduction

Let denote the class of meromorphic functions normalized by which are analytic in the punctured unit disk . For , we denote by and the subclasses of consisting of all meromorphic functions which are, respectively, starlike of order and convex of order in (cf. e.g., [1–4]).

For functions defined by we denote the Hadamard product (or convolution) of and by Let us define the function by for , and , where is the Pochhammer symbol. We note that where is the well-known Gaussian hypergeometric function.

Let us put Corresponding to the functions and and using the Hadamard product for , we define a new linear operator on by The meromorphic functions with the generalized hypergeometric functions were considered recently by Dziok and Srivastava [5, 6], Liu [7], Liu and Srivastava [8–10], and Cho and Kim [11].

For a function , we define and, for , Note that if , , the operator reduced to the one introduced by Cho et al. [12] for . It was known that the definition of the operator was motivated essentially by the Choi-Saigo-Srivastava operator [13] for analytic functions, which includes a simpler integral operator studied earlier by Noor [14] and others (cf. [15–17]). Note also the operator has been recently introduced and studied by Ghanim and Darus [18] and Ghanim et al. [19], respectively. To our best knowledge, the recent work regarding operator was charmingly studied by Piejko and Sokól [20]. Moreover, the operator was then defined and studied by Ghanim and Darus [21]. In the same direction, we will study for the operator given in (1.10).

Now, it follows from (1.8) and (1.10) that Making use of the operator , we say that a function is in the class if it satisfies the following subordination condition: Furthermore, we say that a function is a subclass of the class of the form The main object of this paper is to present several inclusion relations and other properties of functions in the classes and which we have introduced here.

#### 2. Main Results

We begin by recalling the following result (popularly known as Jack's Lemma), which we will apply in proving our first inclusion theorem.

Lemma 2.1 (see [Jack's Lemma] [22]). *Let the (nonconstant) function be analytic in with . If attains its maximum value on the circle at a point , then
**
where is a real number and .*

Theorem 2.2. *If
**
then
*

*Proof. *Let , and suppose that
where the function is either analytic or meromorphic in , with . By using (2.4) and (1.11), we have
Upon differentiating both sides of (2.5) with respect to logarithmically and using the identity (1.11), we obtain
We suppose now that
and apply Jack's Lemma, we thus find that
By writing
and setting in (2.6), we find after some computations that
Set
Then, by hypothesis, we have
which, together, imply that
View of (2.13) and (2.10) would obviously contradict our hypothesis that
Hence, we must have
and we conclude from (2.4) that
The proof of Theorem 2.2 is thus complete.

#### 3. Properties of the Class

Throughout this section, we assume further that and We first determine a necessary and sufficient condition for a function of the form (1.13) to be in the class of meromorphically univalent functions with positive coefficients.

Theorem 3.1. *Let be given by (1.13). Then if and only if
**
where, for convenience, the result is sharp for the function given by
**
for all .*

*Proof. *Suppose that the function is given by (1.13) and is in the class . Then, from (1.13) and (1.12), we find that
Since for any , therefore, we have
Choosing to be real and letting through real values, (3.5) yields
which leads us to the desired inequality (3.2).

Conversely, by applying hypothesis (3.2), we get
Hence, we have . By observing that the function , given by (3.3), is indeed an extremal function for the assertion (3.2), we complete the proof of Theorem 3.1.

By applying Theorem 3.1, we obtain the following sharp coefficient estimates.

Corollary 3.2. *Let be given by (1.13). If , then
**
where the equality holds true for the function given by (3.3).*

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

Theorem 3.3. *If the function defined by (1.13) is in the class , then, for , we have
**
Each of these results is sharp with the extremal function given by (3.3).*

*Proof. *Since . Theorem 3.1 readily yields the inequality
Thus, for and utilizing (3.10), we have

Also from Theorem 3.1, we get
Hence
This completes the proof of Theorem 3.3.

We conclude this section by determining the radii of meromorphically univalent starlikeness and meromorphically univalent convexity of the class . We state our results as in the following theorems.

Theorem 3.4. *Let . Then, is meromorphically univalent starlike of order in , where
**
The equality is attained for the function given by (3.3).*

*Proof. *It suffices to prove that
for , we have
Hence, (3.16) holds true if
or
with the aid of (3.18) and (3.2), it is true to have
Solving (3.19) for , we obtain
This completes the proof of Theorem 3.4.

Theorem 3.5. *Let . Then, is meromorphically univalent convex of order in , where
**
The equality is attained for the function given by (3.3).*

*Proof. *By using the technique employed in the proof of Theorem 3.4, we can show that
for , with the aid of Theorem 3.1. Thus, we have the assertion of Theorem 3.5.

#### Acknowledgment

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

#### References

- M. K. Aouf, “On a certain class of meromorphic univalent functions with positive coefficients,”
*Rendiconti di Matematica e delle sue Applicazioni*, vol. 7, no. 11, pp. 209–219, 1991. View at Google Scholar · View at Zentralblatt MATH - S. K. Bajpai, “A note on a class of meromorphic univalent functions,”
*Revue Roumaine de Mathématiques Pures et Appliquées*, vol. 22, no. 3, pp. 295–297, 1977. View at Google Scholar · View at Zentralblatt MATH - N. E. Cho, S. H. Lee, and S. Owa, “A class of meromorphic univalent functions with positive coefficients,”
*Kobe Journal of Mathematics*, vol. 4, no. 1, pp. 43–50, 1987. View at Google Scholar · View at Zentralblatt MATH - B. A. Uralegaddi and M. D. Ganigi, “A certain class of meromorphically starlike functions with positive coefficients,”
*Pure and Applied Mathematika Sciences*, vol. 26, no. 1-2, pp. 75–81, 1987. View at Google Scholar · View at Zentralblatt MATH - J. Dziok and H. M. Srivastava, “Some subclasses of analytic functions with fixed argument of coefficients associated with the generalized hypergeometric function,”
*Advanced Studies in Contemporary Mathematics, Kyungshang*, vol. 5, no. 2, pp. 115–125, 2002. View at Google Scholar · View at Zentralblatt MATH - J. Dziok and H. M. Srivastava, “Certain subclasses of analytic functions associated with the generalized hypergeometric function,”
*Integral Transforms and Special Functions*, vol. 14, no. 1, pp. 7–18, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-L. Liu, “A linear operator and its applications on meromorphic $p$-valent functions,”
*Bulletin of the Institute of Mathematics. Academia Sinica*, vol. 31, no. 1, pp. 23–32, 2003. View at Google Scholar - 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-L. Liu and H. M. Srivastava, “Certain properties of the Dziok-Srivastava operator,”
*Applied Mathematics and Computation*, vol. 159, no. 2, pp. 485–493, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at 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 · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. E. Cho and I. H. Kim, “Inclusion properties of certain classes of meromorphic functions associated with the generalized hypergeometric function,”
*Applied Mathematics and Computation*, vol. 187, no. 1, pp. 115–121, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. E. Cho, O. S. Kwon, and H. M. Srivastava, “Inclusion and argument properties for certain subclasses of meromorphic functions associated with a family of multiplier transformations,”
*Journal of Mathematical Analysis and Applications*, vol. 300, no. 2, pp. 505–520, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. H. Choi, M. Saigo, and H. M. Srivastava, “Some inclusion properties of a certain family of integral operators,”
*Journal of Mathematical Analysis and Applications*, vol. 276, no. 1, pp. 432–445, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - K. I. Noor, “On new classes of integral operators,”
*Journal of Natural Geometry*, vol. 16, no. 1-2, pp. 71–80, 1999. View at Google Scholar · View at Zentralblatt MATH - K. I. Noor and M. A. Noor, “On Integral Operators,”
*Journal of Natural Geometry*, vol. 238, no. 2, pp. 341–352, 1999. View at Google Scholar - J. Liu, “The Noor integral and strongly starlike functions,”
*Journal of Mathematical Analysis and Applications*, vol. 261, no. 2, pp. 441–447, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J.-L. Liu and K. I. Noor, “Some properties of Noor integral operator,”
*Journal of Natural Geometry*, vol. 21, no. 1-2, pp. 81–90, 2002. View at Google Scholar · View at Zentralblatt MATH - F. Ghanim and M. Darus, “Linear operators associated with a subclass of hypergeometric meromorphic uniformly convex functions,”
*Acta Universitatis Apulensis., Mathematics, Informatics*, no. 17, pp. 49–60, 2009. View at Google Scholar · View at Zentralblatt MATH - F. Ghanim, M. Darus, and A. Swaminathan, “New subclass of hypergeometric meromorphic functions,”
*Far East Journal of Mathematical Sciences*, vol. 34, no. 2, pp. 245–256, 2009. View at Google Scholar · View at Zentralblatt MATH - K. Piejko and J. Sokół, “Subclasses of meromorphic functions associated with the Cho-Kwon-Srivastava operator,”
*Journal of Mathematical Analysis and Applications*, vol. 337, no. 2, pp. 1261–1266, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Ghanim and M. Darus, “Certain subclasses of meromorphic functions related to Cho-Kwon-Srivastava operator,”
*Far East Journal of Mathematical Sciences*, vol. 48, no. 2, pp. 159–173, 2011. View at Google Scholar - I. S. Jack, “Functions starlike and convex of order $\alpha $,”
*Journal of the London Mathematical Society*, vol. 2, no. 3, pp. 469–474, 1971. View at Publisher · View at Google Scholar · View at Zentralblatt MATH