Certain Subclasses of β-Uniformly q-Starlike and β-Uniformly q-Convex Functions

Mathematics Department, Gaza University, State of Palestine Statistics Department, Gaza University, State of Palestine Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia Department of Medical Research, China Medical University, 40402 Taichung, Taiwan Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan Department of Mathematics, Ҫankaya University, 06790 Ankara, Turkey


Introduction
The q-analysis is a generalization of the ordinary analysis. The application of the q-calculus was first introduced by Jackson [1][2][3]. In geometric function theory, the q-hypergeometric functions were first used by Srivastava [4]. The q -calculus provides valuable tools that have been used to define several subclasses of the normalized analytic function in the open unit disk U. Ismail et al. [5] were the first to study a certain class S * of starlike functions by using the q-derivative operator. Recently, new subclasses of analytic functions associated with q-derivative operators are introduced and discussed, see for example [4,[6][7][8][9][10][11][12][13][14][15][16][17][18]. Motivated by the importance of q-analysis, in this paper, we introduce the classes of β -uniformly q-starlike and β-uniformly q-convex functions defined by the q-derivative operator in the open unit disc, as a generalization of β-uniformly starlike and β-uniformly convex functions.
First, we recall some basic notations and definitions from q-calculus, which are used in this paper. The q-derivative of the function f is defined as follows [1][2][3]: From equation (1), it is clear that if f and g are the two functions, then where c is a constant. We note that D q f ðzÞ ⟶ f ′ ðzÞ as q ⟶ 1 − , where f ′ is the ordinary derivative of the function f . In particular, using equation (1), the q-derivative of the function hðzÞ = z n is as follows: where ½n q denotes the q-number and is given as follows: Since we note that ½n q ⟶ n as q ⟶ 1 − , therefore, in view of equation (4), D q hðzÞ ⟶ h′ðzÞ as q ⟶ 1 − , where h ′ ðzÞ denotes the ordinary derivative of the function hðzÞ with respect to z.
In this paper, we consider the classes A and T of the functions, analytic in the open unit disc U = fz ∈ ℂ : jzj < 1g, of the following forms, respectively: Also, using equations (2), (3), (4), and (6), we get the following q-derivatives of the function f : where ½n q is given by equation (5). The classes of starlike functions of order αð0 ≤ α < 1Þ and convex functions of order αð0 ≤ α < 1Þ, denoted by S * ðαÞ and KðαÞ, respectively, are defined as follows [19]: It is clear that S * ðαÞ and KðαÞ are the subclasses of the class A.
The classes of β-uniformly starlike functions of order α and β-uniformly convex functions of order α, denoted by SDðα, βÞ and KDðα, βÞ, respectively, are defined as follows [20]: where z ∈ U, 0 ≤ α < 1, and β ≥ 0: The class of q-starlike functions of order μ, denoted by S * q ðμÞ, is defined as follows [13]: Also, the class of q-convex functions of order μ, denoted by C q ðμÞ, is defined as [13]: The analytic function g is said to be subordinate to the analytic function f in U [21], represented as follows: if there exists a Schwarz function w, which is analytic in U with such that In the next section, we introduce the classes of β-uniformly q-starlike and β-uniformly q-convex functions of order α, denoted by S q ðα, βÞ and UCV q ðα, βÞ, respectively. Also, we obtain the coefficient bounds of the functions belonging to these classes. Definition 1. The function f ∈ A is said to be β-uniformly q -starlike of order α, if it satisfies the following inequality: where 0 < q < 1, β ≥ 0, 0 ≤ α < 1, and z ∈ U.
Also, the relation between the subclasses C q ðαÞ and UCV q ðα, βÞis given by the following result.
Next, the coefficient bound of the class S q ðα, βÞ is given by the following result.
Proof. Now, using the fact that −RðzÞ ≤ jzj, we have
Also, we obtain the coefficient bound for f ∈ T S q ðα, βÞ in the following result.
Proof. Since T is a subclass of class A, therefore in view of Theorem 6, the sufficient condition of our result holds. Now, we need to prove only the necessary condition. Let f ∈ T S q ðα, βÞ and taking z real, then from inequality (19), we have Now, using equations (7) and (8) in inequality (35), we get 1 − ∑ ∞ n=2 n ½ q a n z n−1 1 − ∑ ∞ n=2 a n z n−1 − α > ∑ ∞ n=2 β n ½ q a n z n−1 1 − ∑ ∞ n=2 a n z n−1 , ð36Þ then, letting z → 1 along the real axis, inequality (36), gives the condition (34).
The coefficient bound of the class UCV q ðα, βÞ is given by the following result.
The coefficient bound for f ∈ UCT q ðα, βÞ is given by the following result.
Proof. Since T is a subclass of class A, therefore, in view of Theorem 8, the sufficient condition holds. Now, we need to prove only the necessary condition. Let f belong to the class UCT q ðα, βÞ and taking z real, then from inequality (20), we have Now, using equations (8) and (9) in inequality (44), we get 1 − ∑ ∞ n=2 n ½ 2 q a n z n−1 1 − ∑ ∞ n=2 n ½ q a n z n−1 − α > ∑ ∞ n=2 β n ½ 2 q − n ½ q a n z n−1 then letting z → 1 along real axis, inequality (45) gives condition (43).
We note that, q → 1 − in Theorems 6 and 8, we get the coefficient bounds for the functions belonging to the classes SDðα, βÞ and KDðα, βÞ in [20], respectively.
In the next section, we obtain the extreme points for the functions belonging to the classes T S q ðα, βÞ and UCT q ðα, βÞ.

Extreme Points
The extreme points of f ∈ T S q ðα, βÞ are given by the following result.
Theorem 10. Let f f n ðzÞg n∈ℕ be sequences of functions such that where ½n q denotes the q-number. Then f belongs to T S q ðα, βÞ if and only if f can be expressed as the form where λ n ≥ 0ðn ≥ 1Þ and ∑ ∞ n=1 λ n = 1.
Proof. Let f ∈ T S q ðα, βÞ, then in view of Theorem 7, inequality (34) holds. Since a n ≥ 0ðn ≥ 1Þ and 0 ≤ α < 1, therefore from inequality (34), we have Thus, if we take since λ 1 ≥ 0, then, λ n ≥ 0ðn ≥ 1Þ. Substitutinga n from equation (49) witha n from equation (7), we get: Since ∑ ∞ n=1 λ n = 1, therefore, we have since f 1 ðzÞ = z and f n ðzÞ is given by equation (46). Therefore, from equation (51), we get the assertion (47). Conversely, let f be expressible in the form (47), which on using equation (46), gives 5 Journal of Function Spaces which can be expressed as follows: where Now, to prove that the function f , given by equation (53), belongs to the class T S q ðα, βÞ, we need to show that the coefficients η n ðn ≥ 2Þ satisfy the inequality (34).
Since λ 1 ≥ 0 and ∑ ∞ n=1 λ n = 1, therefore from equation (54), we have Thus, we get Therefore, in view of Theorem 7 and the above inequality, we proved that the function f , given by equation (53), belongs to the class T S q ðα, βÞ.
Also, the extreme points of f ∈ UCT q ðα, βÞ are given by the following result.
Conversely, let f be expressible in the form (47), which on using equation (60), gives which can be expressed as where η n = 1 − α n ½ q n ½ q 1 + β ð Þ− α + β ð Þ λ n , n ≥ 2: Now, to prove that function f is given by equation (63) and belongs to the class UCT q ðα, βÞ, we need to show that the coefficient η n ðn ≥ 2Þ satisfies inequality (43). Since λ 1 ≥ 0 and ∑ ∞ n=1 λ n = 1,, therefore from equation (64), we have Thus, we get Therefore, in view of Theorem 9 and the above inequality, we proved that function f , given by equation (63), belongs to the class ∈UCT q ðα, βÞ.