Journal of Complex Analysis

Volume 2013 (2013), Article ID 754598, 5 pages

http://dx.doi.org/10.1155/2013/754598

Research Article

Inclusion and Neighborhood Properties for Certain Classes of Multivalently Analytic Functions

Serap Bulut

Civil Aviation College, Kocaeli University, Arslanbey Campus, İzmit, 41285 Kocaeli, Turkey

Received 10 May 2013; Accepted 5 October 2013

Academic Editor: Lianzhong Yang

Copyright © 2013 Serap Bulut. 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.

We introduce and investigate two new general subclasses of multivalently analytic functions of complex order by making use of the familiar convolution structure of analytic functions. Among the various results obtained here for each of these function classes, we derive the coefficient inequalities and other interesting properties and characteristics for functions belonging to the classes introduced here.

1. Introduction and Definitions

Let be the set of real numbers, let be the set of complex numbers, let be the set of positive integers, and let .

Let denote the class of functions of the form

which are analytic and -valent in the open unit disk

Denote by the Hadamard product (or convolution) of the functions and ; that is, if is given by (1) and is given by

then

Definition 1. Let the function . Then one says that is in the class if it satisfies the condition

where

is given by (3), and denotes the falling factorial defined as follows:

Various special cases of the class were considered by many earlier researchers on this topic of Geometric Function Theory. For example, reduces to the function class

(i) for , and , studied by Mostafa and Aouf [1];(ii)for and , studied by Srivastava et al. [2];(iii) for , and , studied by Prajapat et al. [3];(iv) for , and , studied by Srivastava and Bulut [4];(v)for , , , and , studied by Ali et al. [5].

Definition 2. Let the function . Then one says that is in the class if it satisfies the condition

where and are defined by (3) and (6), respectively.

Setting , in Definition 2, we have the special class (which generalizes the class defined by Prajapat et al. [3]) introduced by Srivastava et al. [2].

Following a recent investigation by Frasin and Darus [6], if and , then we define the -neighborhood of the function by

It follows from the definition (9) that if

then

The main object of this paper is to investigate the various properties and characteristics of functions belonging to the above-defined classes

Apart from deriving coefficient bounds and coefficient inequalities for each of these classes, we establish several inclusion relationships involving the -neighborhoods of functions belonging to the general classes which are introduced above.

2. Coefficient Bounds and Coefficient Inequalities

We begin by proving a necessary and sufficient condition for the function to be in each of the classes

Theorem 3. Let the function be given by (1). Then is in the class if and only if

where

Proof. We first suppose that the function given by (1) is in the class . Then, in view of (3)–(6), we have

or equivalently

If we choose to be real and let , we arrive easily at the inequality (14).

Conversely, we suppose that the inequality (14) holds true and let

Then, we find that

Hence, by the Maximum Modulus Theorem, we have

which evidently completes the proof of Theorem 3.

Remark 4. If we set and in Theorem 3, then we have [2, Theorem 1].

Lemma 5. Let the function given by (1) be in the class . Then, for , one has

where is defined by (15).

Proof. Let . Then, in view of the assertion (14), we have

Furthermore, by rewriting the assertion (14) as follows:

we obtain

Similar to Theorem 3, we can prove the following result.

Theorem 6. Let the function be given by (1). Then is in the class if and only if

where is defined by (15).

Remark 7. If we set and in Theorem 6, then we have [2, Theorem 2].

Lemma 8. Let the function given by (1) be in the class . Then, for , one has

where is defined by (15).

Proof. Let . Then, in view of the assertion (26), we have

Furthermore, we also have from the assertion (26)

3. A Set of Inclusion Relationships

In this section, we determine inclusion relations for the classes

involving -neighborhoods defined by (9) and (11).

Theorem 9. If and

then

where and are given by (10) and (15), respectively.

Proof. The inclusion relation (33) would follow readily from the definition (11) and the assertion (22).

Remark 10. If we set and in Theorem 9, then we have [2, Theorem 3].

Theorem 11. If and

then

where and are given by (10) and (15), respectively.

Proof. The inclusion relation (35) would follow readily from the definition (11) and the assertion (28).

Remark 12. If we set and in Theorem 11, then we have [2, Theorem 4].

4. Neighborhood Properties

In this section, we determine the neighborhood properties for each of the function classes

which are defined as follows.

Definition 13. A function is said to be in the class if there exists a function such that

Definition 14. A function is said to be in the class if there exists a function such that the inequality (37) holds true.

Setting in Definitions 13 and 14, we have the special classes

introduced by Srivastava et al. [2], respectively.

Theorem 15. If and

then

where is defined by (15).

Proof. Suppose that . Then we find from (9) that

which readily implies that

Since , we find from (21) that

so that

where is given by (39). Thus, by Definition 13, . This completes the proof of Theorem 15.

Remark 16. If we set and in Theorem 15, then we have [2, Theorem 5].

The proof of Theorem 17 (based upon Definition 14) is similar to that of Theorem 15. Therefore we omit the details involved.

Theorem 17. If and

then

where is defined by (15).

Remark 18. If we set and in Theorem 17, then we have [2, Theorem 6].

Acknowledgment

The present investigation was supported by the Kocaeli University under Grant HD 2011/22.

References

A. O. Mostafa and M. K. Aouf, “Neighborhoods of certain -valent analytic functions with complex order,” Computers & Mathematics with Applications, vol. 58, no. 6, pp. 1183–1189, 2009.

H. M. Srivastava, S. S. Eker, and B. Şeker, “Inclusion and neighborhood properties for certain classes of multivalently analytic functions of complex order associated with the convolution structure,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 66–71, 2009.

J. K. Prajapat, R. K. Raina, and H. M. Srivastava, “Inclusion and neighborhood properties for certain classes of multivalently analytic functions associated with the convolution structure,” Journal of Inequalities in Pure and Applied Mathematics, vol. 8, no. 1, article 7, 8 pages, 2007.

H. M. Srivastava and S. Bulut, “Neighborhood properties of certain classes of multivalently analytic functions associated with the convolution structure,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6511–6518, 2012.

R. M. Ali, M. H. Khan, V. Ravichandran, and K. G. Subramanian, “A class of multivalent functions with negative coefficients defined by convolution,” Bulletin of the Korean Mathematical Society, vol. 43, no. 1, pp. 179–188, 2006.

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