Abstract

In this paper, the authors introduced certain subclasses -uniformly -starlike and -uniformly -convex functions of order involving the -derivative operator defined in the open unit disc. Coefficient bounds were also investigated.

1. Introduction

The -analysis is a generalization of the ordinary analysis. The application of the -calculus was first introduced by Jackson [1–3]. In geometric function theory, the -hypergeometric functions were first used by Srivastava [4]. The -calculus provides valuable tools that have been used to define several subclasses of the normalized analytic function in the open unit disk . Ismail et al. [5] were the first to study a certain class of starlike functions by using the -derivative operator. Recently, new subclasses of analytic functions associated with -derivative operators are introduced and discussed, see for example [4, 6–18]. Motivated by the importance of -analysis, in this paper, we introduce the classes of -uniformly -starlike and -uniformly -convex functions defined by the -derivative operator in the open unit disc, as a generalization of -uniformly starlike and -uniformly convex functions.

First, we recall some basic notations and definitions from -calculus, which are used in this paper. The -derivative of the function is defined as follows [1–3]:

From equation (1), it is clear that if and are the two functions, then where is a constant. We note that as , where is the ordinary derivative of the function .

In particular, using equation (1), the -derivative of the function is as follows: where denotes the -number and is given as follows:

Since we note that as , therefore, in view of equation (4), as , where denotes the ordinary derivative of the function with respect to .

In this paper, we consider the classes and of the functions, analytic in the open unit disc , of the following forms, respectively:

Also, using equations (2), (3), (4), and (6), we get the following -derivatives of the function : where is given by equation (5).

The classes of starlike functions of order and convex functions of order , denoted by and , respectively, are defined as follows [19]:

It is clear that and are the subclasses of the class .

The classes of -uniformly starlike functions of order and -uniformly convex functions of order , denoted by and , respectively, are defined as follows [20]: where , , and

The class of -starlike functions of order , denoted by , is defined as follows [13]:

Also, the class of -convex functions of order , denoted by , is defined as [13]:

The analytic function is said to be subordinate to the analytic function in [21], represented as follows: if there exists a Schwarz function , which is analytic in with such that

In the next section, we introduce the classes of -uniformly -starlike and -uniformly -convex functions of order , denoted by and , respectively. Also, we obtain the coefficient bounds of the functions belonging to these classes.

2. Coefficient Bounds

Since the -derivative is a generalized form of the ordinary derivative, therefore, in view of definitions of and , we define the classes of -uniformly -starlike and -uniformly -convex functions of order , denoted by and , respectively, by replacing the ordinary derivative with the -derivative in equations (12) and (13).

We provide the respective definitions of the classes and .

Definition 1. The function is said to be -uniformly -starlike of order , if it satisfies the following inequality: where , , , and .

Definition 2. The function is said to be -uniformly -convex of order , if it satisfies the following inequality: where , , , and .

Further, we define the classes and containing functions with negative coefficients and satisfying inequalities (19) and (20), respectively, as follows:

Remark 3. We note that (i) and (ii) and (see [8]).(iii) and

Now, the relation between the subclasses and is given by the following result.

Theorem 4. Let , then , where , , and .

Proof. If , then in view of Definition 1 and using the fact that , we get which implies that then Since and , then . Hence, in view of equation (14), we obtain .

Also, the relation between the subclasses and is given by the following result.

Theorem 5. Let , then , where , , and .

Proof. If , then in view of Definition 2 and using the fact that , we get which implies that then since and , then . Hence, in view of equation (15), we obtain .

Next, the coefficient bound of the class is given by the following result.

Theorem 6. A function belongs to the class if where , , , and denotes the -number.

Proof. Now, using the fact that , we have Using equations (6) and (8) in the right hand side of inequality (29), we get Since , therefore, from the above inequality, we get Combining inequalities (29) and (31), we get If , which is equivalent to inequality (28), then from inequality (32) we get which is equivalent to inequality (19). Thus, in view of Definition 1, the function .

Also, we obtain the coefficient bound for in the following result.

Theorem 7. The function belongs to the class , if and only if where , , , and denotes the -number.

Proof. Since is a subclass of class , therefore in view of Theorem 6, the sufficient condition of our result holds. Now, we need to prove only the necessary condition. Let and taking real, then from inequality (19), we have Now, using equations (7) and (8) in inequality (35), we get then, letting along the real axis, inequality (36), gives the condition (34).

The coefficient bound of the class is given by the following result.

Theorem 8. A function belongs to the class if where , , , and denotes the -number.

Proof. Now, using the fact that , we have Using equations (8) and (9) in the right hand side of inequality (38), we get Since , therefore, from the above inequality, we get Combining inequalities (38) and (40), we get If , which is equivalent to inequality (37), then from inequality (41), we get which is equivalent to inequality (20). Thus, in view of Definition 2, the function .