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 [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