Inclusion and Neighborhood Properties for Certain Classes of Multivalently Analytic Functions
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 ;(ii)for and , studied by Srivastava et al. ;(iii) for , and , studied by Prajapat et al. ;(iv) for , and , studied by Srivastava and Bulut ;(v)for , , , and , studied by Ali et al. .
Following a recent investigation by Frasin and Darus , 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
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.
Similar to Theorem 3, we can prove the following result.
3. A Set of Inclusion Relationships
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.
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.
Theorem 17. If and then where is defined by (15).
The present investigation was supported by the Kocaeli University under Grant HD 2011/22.
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.View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
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.View at: Google Scholar | Zentralblatt MATH | MathSciNet
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