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The Scientific World Journal

Volume 2013 (2013), Article ID 958796, 6 pages

http://dx.doi.org/10.1155/2013/958796

## Certain Subclasses of Analytic Functions with Complex Order

^{1}Department of Mathematics, VHNSN College, Virudhunagar 626001, India^{2}Department of Mathematics, Government Arts College, Melur 625106, India

Received 12 August 2013; Accepted 12 September 2013

Academic Editors: G. Bonanno and A. Ibeas

Copyright © 2013 A. Selvam et al. 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

Two new subclasses of analytic functions of complex order are introduced. Apart from establishing coefficient bounds for these classes, we establish inclusion relationships involving () neighborhoods of analytic functions with negative coefficients belonging to these subclasses.

#### 1. Introduction

Let denote the class of functions of the form which are analytic and univalent in the open disc

A function is star-like of complex order , denoted as if and only if it satisfies

A function is convex of complex order , denoted as if and only if it satisfies These classes, , and are introduced and studied by Nasr and Aouf [1] and Wiatrowski [2].

For the two functions given by the Hadamard product or convolution, denoted by , is given by

Given of the form (1) and , we define neighborhood of a function as In particular, for the identity function ,

The concept of Neighborhood of a function is introduced and studied by Ruscheweyh [3] and extended further by Silverman [4].

For complex numbers and (; for ), we define the generalized hypergeometric function as where denotes the set of all positive integers and is the Pochhammer symbol defined in terms of gamma functions as Corresponding to the function defined by recently in [5], an operator is defined by where and . By the above definition, it is easy to note that Let us take for convenience that Hence, we have For suitable values of , , , , , , and , we can deduce several operators such as Sălăgean derivative operator [6], Ruscheweyh derivative operator [7], fractional calculus operator [8], Carlson-Shaffer operator [9], Dziok-Srivatsava operator [10], and also the operator introduced by Abubaker and Darus [11].

*Definition 1. *For , we let be the subclass of consisting of functions of the form (1) that satisfy
where , , , and are as given in (15).

*Definition 2. *For we let be the subclass of , consisting of functions of the form (1) that satisfy
where , , , and are as given in (15).

By specializing the parameters involved in the above definitions, we could arrive at several known as well as new classes. For example, by taking , , , , , and and the above classes reduced to where denote the Sălăgean derivative of order given by

Similarly, on taking , , , , , , one gets where is the operator introduced and studied by Abubaker and Darus [11] given by

Further, by taking in the definition of the classes and , we could arrive at and which were introduced and studied by Murugusundaramoorthy et al. [12].

In this paper, we establish the coefficient inequalities for the classes and and several inclusion relationships involving neighborhoods of analytic univalent functions with negative and missing coefficients belonging to these classes.

#### 2. Coefficient Inequalities

Theorem 3. *Let the function as given in (1). Then, if and only if
*

*Proof. *Let the functions of form (1) belong to the class . Then, in view of (15) and (16), we get
Letting through real values, we get the required assertion (22). Conversely, suppose satisfies (22), then, in view of (16), consider
Hence, the result follows.

Similarly, we prove the following.

Theorem 4. *Let the function be as defined in (1). Then, if and only if
*

Corollary 5. *Let the function as given in (1). Then, if and only if
*

Corollary 6. *Let the function be as defined in (1). Then, if and only if
*

Corollary 7. *Let the function as given in (1). Then, if and only if
*

Corollary 8. *Let the function be as defined in (1). Then, if and only if
*

#### 3. Inclusion Relationships

Theorem 9. *If
**
then .*

*Proof. *Let . Then, in view of (22), we have
Consider
Hence,
Hence, the result follows.

In similar manner, we establish the following result.

Theorem 10. *If
**
then .*

Corollary 11. *If
**
then .*

Corollary 12. *If
**
then .*

Corollary 13. *If
**
then . *

Corollary 14. *If
**
then . *

#### 4. Neighborhoods for and

In this section, we determine the neighborhood properties of and . Here, the classes consist of functions for which there exists a function such that In the same way, we define , consisting of functions for which there exists another function such that

Theorem 15. *If and
**
then .*

*Proof. *Suppose , then

Since , we have
Consider
Therefore, for given by (42).

Theorem 16. *If and
**
then . *

Corollary 17. *If and
**
then . *

Corollary 18. *If and
**
then . *

Corollary 19. *If and
**
then . *

Corollary 20. *If and
**
then .*

#### Conflict of Interests

The authors declare that they do not have conflict of interests regarding the publication of this paper.

#### References

- M. Nasr and M. Aouf, “Starlike functions of complex order,”
*Journal of Natural Sciences and Mathematics*, vol. 25, no. 1, pp. 1–12, 1985. - P. Wiatrowski, “On the coefficients of some family of holomorphic functions,”
*Zeszyty Naukowe*, vol. 2, no. 39, pp. 75–85, 1970. - St. Ruscheweyh, “Neighborhoods of univalent functions,”
*Proceedings of the American Mathematical Society*, pp. 521–527, 1981. - H. Silverman, “Neighborhoods of a class of analytic functions,”
*Far East Journal of Mathematical Sciences*, vol. 3, no. 2, pp. 165–169, 1995. - N. Marikkannan, “A subclass of analytic functions and a generalised differential operator,” communicated.
- G. S. Salagean, “Subclasses of univalent functions,” in
*Complex Analysis*, vol. 1013 of*Lecture Notes in Mathematics*, pp. 362–372, Berlin, Germany, 1981, Proceedings of the 5th Romanian-Finnish Seminar, Part 1, Bucharest, Romania, 1981. - St. Ruscheweyh, “New criteria for univalent functions,”
*Proceeding of the American Mathematical Society*, vol. 49, pp. 109–115, 1975. - S. Owa, “Some applications of the fractional calculus,” in
*Proceedings of the Workshop on Fractional Calculus*, vol. 138 of*Research Notes in Mathematics*, pp. 164–175, University of Strathclyde, Ross Priory, UK, 1985. - B. C. Carlson and S. B. Shaffer, “Starlike and Prestarlike hypergeometric functions,”
*SIAM Journal on Mathematical Analysis*, vol. 15, no. 4, pp. 737–745, 2002. - J. Dziok and H. M. Srivatsava, “Classes of analytic functions associated with the generalized hypergeometric function,”
*Applied Mathematics and Computation*, vol. 103, no. 1, pp. 1–13, 1999. - A. A. A. Abubaker and M. Darus, “Neighborhoods of certain classes of analytic functions defined by a generalized differential operator,”
*International Journal of Mathematical Analysis*, vol. 4, no. 45-48, pp. 2373–2380, 2010. View at Scopus - G. Murugusundaramoorthy, T. Rosy, and S. Sivasubramanian, “On certain classes of analytic functions
of complex order defined by Dziok-Srivatsava operator,”
*Journal of Mathematical Inequalities*, 2007. View at Publisher · View at Google Scholar