The Scientific World Journal

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

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

Research Article

Certain Subclasses of Analytic Functions with Complex Order

A. Selvam,¹ P. Sooriya Kala,¹ and N. Marikkannan²

¹Department of Mathematics, VHNSN College, Virudhunagar 626001, India

²Department 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.

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