Subclasses of Analytic Functions Associated with Generalised Multiplier Transformations
Rashidah Omar1,2and Suzeini Abdul Halim2
Academic Editor: W. Yu, O. Miyagaki
Received20 Jan 2012
Accepted25 Mar 2012
Published24 May 2012
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 , , , CΗtaΕ [1] defined the multiplier transformations by the following series:
Recently, some properties of functions using the multiplier transformations have been studied in [2β6]. 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 CtaΕ 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 [9β11] 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 [21β24]
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. CΗtaΕ, βOn certain classes of p-valent functions defined by new multiplier transformations,β TC Istanbul Kultur University Publications, TC Istanbul kultur University, vol. 91, pp. 241β250, 2008, Proceedings of the International Symposium on Geometric Function Theory and Applications (GFTA '07), Istanbul, Turkey, August 2007.
A. CΗtaΕ, G. I. Oros, and G. Oros, βDifferential subordinations associated with multiplier transformations,β Abstract and Applied Analysis, vol. 2008, Article ID 845724, 11 pages, 2008.
A. CΗtaΕ, βNeighborhoods of a certain class of analytic functions with negative coefficients,β Banach Journal of Mathematical Analysis, vol. 3, no. 1, pp. 111β121, 2009.
N. E. Cho and K. I. Noor, βSandwich-type theorems for a class of multiplier transformations associated with the Noor integral operators,β Abstract and Applied Analysis, vol. 2012, Article ID 904272, 13 pages, 2012.
R. M. El-Ashwah, M. K. Aouf, and S. M. El-Deeb, βOn a class of multivalent functions defined by an extended multiplier transformations,β Computers and Mathematics with Applications, vol. 60, no. 3, pp. 623β628, 2010.
A. A. LupaΕ, βA note on a subclass of analytic functions defined by Ruscheweyh derivative and multiplier transformations,β International Journal of Open Problems in Complex Analysis, vol. 2, no. 2, pp. 60β66, 2010.
S. R. Mondal and A. Swaminathan, βGeometric properties of generalized polylogarithm,β Integral Transforms and Special Functions, vol. 21, no. 9, pp. 691β701, 2010.
N. E. Cho and J. A. Kim, βInclusion properties of certain subclasses of analytic functions defined by a multiplier transformation,β Computers and Mathematics with Applications, vol. 52, no. 3-4, pp. 323β330, 2006.
J. H. Choi, M. Saigo, and H. M. Srivastava, βSome inclusion properties of a certain family of integral operators,β Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 432β445, 2002.
O. S. Kwon and N. E. Cho, βInclusion properties for certain subclasses of analytic functions associated with the Dziok-Srivastava operator,β Journal of Inequalities and Applications, vol. 2007, Article ID 51079, 10 pages, 2007.
N. E. Cho and H. M. Srivastava, βArgument estimates of certain analytic functions defined by a class of multiplier transformations,β Mathematical and Computer Modelling, vol. 37, no. 1-2, pp. 39β49, 2003.
F. M. Al-Oboudi, βOn univalent functions defined by derivative operator,β International Journal of Mathematics and Mathematical Sciences, vol. 27, pp. 1429β1436, 2004.
G. S. Salagean, βSubclasses of univalent functions,β in Proceedings of the Complex Analysis 5th Romanian-Finnish Seminar, Part 1, vol. 1013, pp. 362β372, Springer, 1983.
B. A. Uralegaddi and C. Somanatha, βCertain classes of univalent functions,β in Current Topics in Analytic Function Theory, pp. 371β374, World Scientific, River Edge, NJ, USA, 1992.
W. Ma and D. Minda, βA unified treatment of some special classes of univalent functions,β in Proceedings of the Conference on Complex Analysis, Z. Li, F. Ren, L. Yang, and S. Zhang, Eds., pp. 157β169, International Press, Cambridge, Mass, USA, 1992.
P. Enigenberg, S. S. Miller, P. T. Mocanu, and M. O. Reade, βOn a Briot-Bouquet differential subordination,β General Inequalities, vol. 3, pp. 339β348, 1983.
S. S. Miller and P. T. Mocanu, βDifferential subordination and univalent functions,β The Michigan Mathematical Journal, vol. 28, pp. 157β171, 1981.
I. B. Jung, Y. C. Kim, and H. M. Srivastava, βThe Hardy space of analytic functions associated with certain one-parameter families of integral operators,β Journal of Mathematical Analysis and Applications, vol. 176, no. 1, pp. 138β147, 1993.
A. E. Livington, βOn the radius of univalence of certain analytic functions,β Proceedings of the American Mathematical Society, vol. 17, pp. 352β357, 1966.