Research Article | Open Access

# Certain Subclasses of Analytic Functions with Complex Order

**Academic Editor:**G. Bonanno

#### 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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#### Copyright

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.