#### 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 .

The coefficient bound for is given by the following result.

Theorem 9. *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 8, the sufficient condition holds. Now, we need to prove only the necessary condition. Let belong to the class and taking real, then from inequality (20), we have
Now, using equations (8) and (9) in inequality (44), we get
then letting along real axis, inequality (45) gives condition (43).

We note that, in Theorems 6 and 8, we get the coefficient bounds for the functions belonging to the classes and in [20], respectively.

In the next section, we obtain the extreme points for the functions belonging to the classes and .

#### 3. Extreme Points

The extreme points of are given by the following result.

Theorem 10. *Let be sequences of functions such that
where denotes the -number. Then belongs to if and only if can be expressed as the form
where and .*

*Proof. *Let , then in view of Theorem 7, inequality (34) holds. Since and , therefore from inequality (34), we have
Thus, if we take
since , then, .

Substitutingfrom equation (49) withfrom equation (7), we get:
Since , therefore, we have
since and is given by equation (46). Therefore, from equation (51), we get the assertion (47). Conversely, let be expressible in the form (47), which on using equation (46), gives
which can be expressed as follows:
where
Now, to prove that the function , given by equation (53), belongs to the class , we need to show that the coefficients satisfy the inequality (34).

Since and , therefore from equation (54), we have
Thus, we get
Therefore, in view of Theorem 7 and the above inequality, we proved that the function , given by equation (53), belongs to the class .

Also, the extreme points of are given by the following result.

Theorem 11. *Let be a sequence of functions such that
where , , , and . Then, belongs to if and only if can be expressed in the form given by equation (47) in terms of functions , given by equation (57), and , .*

*Proof. *Let , then from inequality (43), we have
If we set
since , then . Then, substituting from equation (59) with equation (7), we get
Since , therefore, we have
since and is given by equation (57). Therefore, from equation (61), we get assertion (47).

Conversely, let be expressible in the form (47), which on using equation (60), gives
which can be expressed as
where
Now, to prove that function is given by equation (63) and belongs to the class , we need to show that the coefficient satisfies inequality (43). Since and , therefore from equation (64), we have
Thus, we get
Therefore, in view of Theorem 9 and the above inequality, we proved that function , given by equation (63), belongs to the class .

#### 4. Partial Sums

The sequence of partial sums of the function , is defined as [22].

Now, we find the bounds of the real part of the ratio of the complex valued function to its partial sums , for the function to be in the class in the following result.

Theorem 12. *Let in the form (6) and suppose that
where
then . Further, the following inequalities hold:
where
*

*Proof. *Since is increasing and , , therefore, in view of equation (69), is an increasing sequence. Then, , and
Since , therefore, we have
Thus, for the particular value of , condition (72) holds. In view of the first inequality of condition (72), we have
which in view of inequality (68), gives
or, equivalently
Now, for some fixed positive integer , we define
Now, using equations (6) and (67), equation (78) gives
From equation (79), we have
In view of inequality (77), the above inequality gives , which implies
Since each , therefore, using equation (79) in inequality (81), we get assertion (70).

Again, since is an increasing function and , , therefore, we have
which in view of inequality (68), gives
Now, we define the function as follows:
Using equations (6) and (67) in equation (84), we get
From equation (85), we have
using inequality (83) in inequality (86), we get , which implies
Therefore, using equation (84) in inequality (87), we get assertion (71).

Now, we find the bounds of the real part of the ratio of the complex valued function to its partial sums , for the function to be in the class in the following result.

Theorem 13. *Let be in the form given by equation (6) and
where
*

Then, . Further, the following inequalities hold: where

*Proof. *Using Theorem 6 and following the same steps involved in the proof of Theorem 12, we get assertion (90) and (91).

In the next section, we discuss the integral means inequality for the functions belonging to the classes and .

#### 5. Integral Means Inequality

Silverman [23] has been using the subordination principle to show that the integral attains its maximum value in class , when . Then, he applied that principle to solve the integral means inequality . Also, he found the integral means inequality for the classes and with negative coefficients.

First, we need to mention the following lemma [24].

Lemma 14. *If and are two analytic functions in in the form and , then
where , , and .*

Now, we establish the integral means inequality for the functions belonging to the class .

Theorem 15. *Let be of the form given by equation (7) that belongs to the class and be defined as follows:
then, for , we have
*

*Proof. *We define the function as follows:
From the above equation, we have
Again, from equation (96), we have
since implies , and using inequality (37), therefore, from the above inequality, we have
From equation (96), we have
Since is analytic in , therefore in view of equations (18), (96), (97), and (100); inequality (99); and the subordination principle, we have
Since, the function on the both sides of the above relation are analytic in , therefore, in view of Lemma 14 and equation (94), we get assertion (95).

Next, we establish the integral means inequality for the functions belonging to the class with the positive coefficients.

Theorem 16. *Let belong to the class and is defined by
then, for , we have
*

*Proof. *We define the function as follows:
From the above equation, we have
Again, from equation (104), we have
since , then and using inequality (103), therefore, from the above inequality, we have
From equation (104)