ISRN Mathematical AnalysisVolume 2012 (2012), Article ID 632429, 9 pagesdoi:10.5402/2012/632429

Research Article

Subclasses of Analytic Functions Associated with Generalised Multiplier Transformations

Rashidah Omar1,2 and Suzeini Abdul Halim²

¹Faculty of Computer and Mathematical Sciences, MARA University of Technology, 40450 Shah Alam, Selangor, Malaysia²Institute of Mathematical Sciences, Faculty of Science, University of Malaya, 50603 Kuala Lumpur, Malaysia

Received 20 January 2012; Accepted 25 March 2012

Academic Editors: O. Miyagaki and W. Yu

Copyright © 2012 Rashidah Omar and Suzeini Abdul Halim. 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.

New subclasses of analytic functions in the open unit disc are introduced which are defined using generalised multiplier transformations. Inclusion theorems are investigated for functions to be in the classes. Furthermore, generalised Bernardi-Libera-Livington integral operator is shown to be preserved for these classes.

1. Introduction

Let denote the class of functions normalised by in the open unit disk . Also let , and denote, respectively, the subclasses of consisting of functions which are starlike, convex, and close to convex in . An analytic function is subordinate to an analytic function , written if there exists an analytic function in such that and for and . In particular, if is univalent in , then is equivalent to and . The convolution of two analytic functions and is defined by .

For any real numbers and where , , , Cǎtaş [1] defined the multiplier transformations by the following series: Recently, some properties of functions using the multiplier transformations have been studied in [2–6]. Using the convolution, we extend the multiplier transformation in (1.1) to be a unified operator. The approach used is similar to Noor's [7], only we generalise and extend to include powers and uses the multiplier Cǎtaş as basis instead of the Ruscheweyh operator.

Set the function and note that, for , is the generalised polylogarithm functions discussed in [8]. A new function is defined in terms of the Hadamard product (or convolution) as follows: Motivated by [9–11] and analogous to (1.1), the following operator is introduced: The operator unifies other previously defined operators. For examples,(i) is the given in [1],(ii) is the given in [12],

also, for any integer ,(iii) given in [13],(iv) given in [14],(v) given in [15].

The following relations are easily derived using the following definition: Let be the class of all analytic and univalent functions in and for which is convex with and for . For , Ma and Minda [16] studied the subclasses , and of the class . These classes are defined using the principle of subordination as follows: Obviously, we have the following relationships for special choices and : Using the generalised multiplier transformations , new classes , and are introduced and defined below It can be shown easily that Janowski [17] introduced class and in particular for , we set In [18], the authors studied the inclusion properties for classes defined using Dziok-Srivastava operator. This paper investigates the similar properties for analytic functions in the classes defined by the generalised multiplier transformations . Furthermore, applications of other families of integral operators are considered involving these classes.

2. Inclusion Properties Involving

In proving our results, the following lemmas are needed.

Lemma 2.1 (see [19]). Let be convex univalent in , with and . If is analytic in with , then

Lemma 2.2 (see [20]). Let be convex univalent in and be analytic in with . If is analytic in and , then

Theorem 2.3. For any real numbers and where and . Let and , then .

Proof. Let , and set where is analytic in with . Rearranging (1.5), we have Next, differentiating (2.3) and multiplying by gives Since and applying Lemma 2.1, it follows that . Thus .

Theorem 2.4. Let , and . Then .

Proof. Let , and from (1.6), we obtain that Making use of the differentiation on both sides in (2.5) and setting , we get the following: Since and , using Lemma 2.1, we conclude that .

Corollary 2.5. Let , and . Then and .

Theorem 2.6. Let and . Then and .

Proof. Using (1.10) and Theorem 2.3, we observe that To prove the second part of theorem, using the similar manner and applying Theorem 2.4, the result is obtained.

Theorem 2.7. Let and . Then and .

Proof. Let . In view of the definition of the class , there is a function such that Applying Theorem 2.3, then and let .Let the analytic function with as Thus, rearranging and differentiating (2.9), we have Making use (1.5), (2.9), (2.10), and , we obtain that Since and , then . Using Lemma 2.2, we conclude that and thus . By using similar manner and (1.6), we obtain the second result.

In summary, using subordination technique inclusion properties has been established for certain analytic functions defined via the generalised multiplier transformation.

3. Inclusion Properties Involving

In this section, we determine properties of generalised Bernardi-Libera-Livington integral operator defined by [21–24] and satisfies the following:

Theorem 3.1. If , then .

Proof. Let , then . Taking the differentiation on both sides of (3.2) and multiplying by , we obtain Setting , we have Lemma 2.1 implies . Hence .

Theorem 3.2. Let , then .

Proof. By using (1.10) and Theorem 3.1, it follows that

Theorem 3.3. Let , and , then .

Proof. Let , then there exists function such that . Since therefore from Theorem 3.1, . Then let Set By rearranging and differentiating (3.7), we obtain that Making use (3.2), (3.7), and (3.6), it can be derived that Hence, applying Lemma 2.2, we conclude that , and it follows that .

For analytic functions in the classes defined by generalised multiplier transformations, the generalised Bernardi-Libera-Livington integral operator has been shown to be preserved in these classes.

4. Conclusion

Results involving functions defined using the generalised multiplier transformation, namely, inclusion properties and the Bernardi-Libera-Livington integral operator were obtained using subordination principles. In [18], similar results were discussed for functions defined using the Dziok-Srivastava operator.

Acknowledgment

This research was supported by IPPP/UPGP/geran(RU/PPP)/PS207/2009A University Malaya Grants 2009.

References

A. Cǎtaş, “On certain classes of p-valent functions defined by new multiplier transformations,” TC Istanbul Kultur University Publications, TC Istanbul kultur University, vol. 91, pp. 241–250, 2008, Proceedings of the International Symposium on Geometric Function Theory and Applications (GFTA '07), Istanbul, Turkey, August 2007

A. Cǎtaş, G. I. Oros, and G. Oros, “Differential subordinations associated with multiplier transformations,” Abstract and Applied Analysis, vol. 2008, Article ID 845724, 11 pages, 2008.

A. Cǎtaş, “Neighborhoods of a certain class of analytic functions with negative coefficients,” Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 111–121, 2009.

N. E. Cho and K. I. Noor, “Sandwich-type theorems for a class of multiplier transformations associated with the Noor integral operators,” Abstract and Applied Analysis, vol. 2012, Article ID 904272, 13 pages, 2012.

R. M. El-Ashwah, M. K. Aouf, and S. M. El-Deeb, “On a class of multivalent functions defined by an extended multiplier transformations,” Computers and Mathematics with Applications, vol. 60, no. 3, pp. 623–628, 2010.

A. A. Lupaş, “A note on a subclass of analytic functions defined by Ruscheweyh derivative and multiplier transformations,” International Journal of Open Problems in Complex Analysis, vol. 2, no. 2, pp. 60–66, 2010.

K. I. Noor, “On new classes of integral operators,” Journal of Natural Geometry, vol. 16, pp. 71–80, 1999.

S. R. Mondal and A. Swaminathan, “Geometric properties of generalized polylogarithm,” Integral Transforms and Special Functions, vol. 21, no. 9, pp. 691–701, 2010.

N. E. Cho and J. A. Kim, “Inclusion properties of certain subclasses of analytic functions defined by a multiplier transformation,” Computers and Mathematics with Applications, vol. 52, no. 3-4, pp. 323–330, 2006.

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.

O. S. Kwon and N. E. Cho, “Inclusion properties for certain subclasses of analytic functions associated with the Dziok-Srivastava operator,” Journal of Inequalities and Applications, vol. 2007, Article ID 51079, 10 pages, 2007.

N. E. Cho and H. M. Srivastava, “Argument estimates of certain analytic functions defined by a class of multiplier transformations,” Mathematical and Computer Modelling, vol. 37, no. 1-2, pp. 39–49, 2003.

F. M. Al-Oboudi, “On univalent functions defined by derivative operator,” International Journal of Mathematics and Mathematical Sciences, vol. 27, pp. 1429–1436, 2004.

G. S. Salagean, “Subclasses of univalent functions,” in Proceedings of the Complex Analysis 5th Romanian-Finnish Seminar, Part 1, vol. 1013, pp. 362–372, Springer, 1983.

B. A. Uralegaddi and C. Somanatha, “Certain classes of univalent functions,” in Current Topics in Analytic Function Theory, pp. 371–374, World Scientific, River Edge, NJ, USA, 1992.

W. Ma and D. Minda, “A unified treatment of some special classes of univalent functions,” in Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang, Eds., pp. 157–169, International Press, Cambridge, Mass, USA, 1992.

W. Janowski, “Some extremal problems for certain families of analytic functions I,” Annales Polonici Mathematici, vol. 28, pp. 297–326, 1973.

R. Omar and S. A. Halim, “Classes of functions defined by Dziok-Srivastavaoperator,” Far East Journal of Mathematical Sciences. In press

P. Enigenberg, S. S. Miller, P. T. Mocanu, and M. O. Reade, “On a Briot-Bouquet differential subordination,” General Inequalities, vol. 3, pp. 339–348, 1983.

S. S. Miller and P. T. Mocanu, “Differential subordination and univalent functions,” The Michigan Mathematical Journal, vol. 28, pp. 157–171, 1981.

S. D. Bernardi, “Convex and starlike univalent functions,” Transactions of the American Mathematical Society, vol. 135, pp. 429–446, 1969.

I. B. Jung, Y. C. Kim, and H. M. Srivastava, “The Hardy space of analytic functions associated with certain one-parameter families of integral operators,” Journal of Mathematical Analysis and Applications, vol. 176, no. 1, pp. 138–147, 1993.

R. J. Libera, “Some classes of regular univalent functions,” Proceedings of the American Mathematical Society, vol. 16, pp. 755–758, 1965.

A. E. Livington, “On the radius of univalence of certain analytic functions,” Proceedings of the American Mathematical Society, vol. 17, pp. 352–357, 1966.