Fekete-Szegö Type Problems and Their Applications for a Subclass of q -Starlike Functions with Respect to Symmetrical Points

: In this article, by using the concept of the quantum (or q -) calculus and a general conic domain Ω k , q , we study a new subclass of normalized analytic functions with respect to symmetrical points in an open unit disk. We solve the Fekete-Szegö type problems for this newly-deﬁned subclass of analytic functions. We also discuss some applications of the main results by using a q -Bernardi integral operator.


Introduction and Definitions
Let A denote the class of all functions f which are analytic in the open unit disk E = {z : z ∈ C and |z| < 1} and has the normalized Taylor-Maclaurin series expansion of the following form: f (z) = z + ∞ ∑ n=2 a n z n .
(1) Let S be the subclass of all functions in A that are univalent in E (see [1]): If f and g ∈ A, the function f is said to be subordinate to the function g, written as f ≺ g, if there exists an analytic function w in E, with w(0) = 0 and |w(z)| < 1 (z ∈ E), such that f (z) = g w(z) . Furthermore, the following equivalence will hold true (see [2]), if g is univalent in E.
Let P denote the well-known Carathéodory class of functions p, which are analytic in the open unit disk E with p(z) > 0 and p(0) = 1.
If p ∈ P, then it has the form given by where |c n | 2 (n ∈ N). If f is univalent in E and f (E) is a star-shaped domain with respect to the origin, then f is called starlike in E with respect to the origin. The analytical condition of a starlike function in E is given as follows: The class of all such functions is denoted by S * . A function f ∈ A is said to be starlike with respect to symmetrical points (see [3]) if it satisfies the inequality: The class of all functions in S which are starlike with respect to symmetrical points is denoted by S * s . Furthermore, we denote two interesting subclasses of S by k-U CV and k-S T (0 k < ∞) of functions which are, respectively, k-uniformly convex and k-starlike in E defined for 0 k < ∞ by Kanas et al. (see [4,5]; see also [6]) defined and studied classes of k-starlike functions and k-uniformly convex functions subject to the conic domain Ω k (k 0), where For this conic domain, the following functions play the role of extremal functions: where K(i) is the first kind of Legendre's complete elliptic integral (see, for details [4,5]). Indeed, from (4), we have p k (z) = 1 + P 1 z + P 2 z 2 + P 3 z 3 + . . .
The quantum (or q-) calculus is an important tool which is used to study various families of analytic functions and has inspired the researchers due to its applications in mathematics and some other related disciplines. Srivastava (see, for details [7]) was the first who used the basic (or q-) hypergeometric functions in Geometric Functions Theory. The extension of the class of starlike functions in the quantum (or q-) calculus was first introduced in [8] by means of the q-difference operator. After that, some remarkable research work was conducted by many mathematicians, which has played an important role in Geometric Function Theory. In particular, Srivastava et al. [9,10] studied the class of q-starlike functions related with the Janowski functions. Mahmood et al. [11] studied the class of q-starlike functions associated with conic regions. The upper bound of the third Hankel determinant for a class of q-starlike functions was investigated in [12] (see also [9]). Kanas and Raducanu [13] introduced the q-analogue of the Ruscheweyh operator by using the concept of convolution and studied some of its properties (see also [11,[14][15][16][17][18][19][20]). Many other q-derivative and q-integral operators can be written by using the idea of convolution (we refer, for details, to [21][22][23][24]). For a comprehensive review of the quantum (or q-)-calculus literature, we refer to a recently-published survey-cum-expository review article by Srivastava [25]. In this article, we will use the conic domain Ω k,q and the quantum (or q-) calculus to define and investigate new subclasses of starlike functions with respect to symmetrical points in the open unit disk E. We will investigate the Hankel determinant, the Toeplitz matrices and the Fekete-Szegö inequalities, and discuss some applications of the main results by using the q-Bernardi integral operator.
We first give some basic definitions of the quantum (or q-) calculus that will help us in the upcoming sections. We also provide some notations and concepts used in this investigation. Definition 1. Let q ∈ (0, 1) and the q-factorial [n] q ! be defined as follows: Definition 2. The generalized q-Pochhammer symbol [t] n,q (t ∈ C) is defined as follows: Definition 3. The q-Gamma function is defined as follows:

Definition 4.
(see [26]) For f ∈ A, the q-derivative operator or q-difference operator are defined as follows: From (1) and (7), we have [n] q a n z n−1 .
Moreover, for n ∈ N and z ∈ E, we get D q z n = [n] q z n−1 , D q ∞ ∑ n=1 a n z n = ∞ ∑ n=1 [n] q a n z n−1 .
When q → 1−, the q-difference operator D q approaches the ordinary differential operator: Definition 5. (see [8]) We say that a function f ∈ A belongs to the class S * q if and By applying the principle of subordination, the conditions (9) and (10) can be written as follows (see [27]): Now, making use of the quantum (or q-) calculus and the principle of subordination, we define q-starlike and q-convex functions with respect to symmetrical points as follows.

Definition 6. An analytic function f is said to be in the class
By applying the principle of subordination, the condition (11) can be written as follows: 2z Definition 7. (see [9]) Let k ∈ [0, ∞) and q ∈ (0, 1) . A function p is said to be in the class k-P q if and only if where and p k (z) is given by (5).
Geometrically, a function p ∈ k-P q takes on all values from the domain Ω k,q , which is defined as follows:
In the present investigation, by using the quantum (or q-) calculus and the general conic domain Ω k,q , we focus on the Hankel determinant, the Toeplitz matrices and the Fekete-Szegö problems for the function class S * s (q).

Definition 8. An analytic function f is said to be in the class k-S
Special Case: For k = 0 and q → 1−, then the class k-S * s (q) reduces to S * s (see [3]).
The symmetric Toeplitz determinant T j (n) is defined as follows: so that and so on. The problem of finding the best possible bounds for ||a n+1 | − |a n || has a long history (see [33]). It is known from [33] that for a constant c.

Lemma 1.
(see [31]) If p is analytic in E and of the form (2), then and and, for some x, z ∈ C, with |z| 1, and |x| 1.
The above inequality is sharp for the function f given by Lemma 4. (see [35]) If p is analytic in E and of the form (2), then The equality holds true for the function p given by p(z) = 1 + z 1 − z or by one of its rotations, when v < 0 or v > 1. In addition, the equality holds true for the function p given by or by one of its rotations, when 0 < v < 1. if v = 0, the equality holds true if and only if or one of its rotations. If v = 1, the equality holds true if and only if p(z) is the reciprocal of one of the functions such that the equality holds true in the case when v = 0. In addition, the above upper bound is sharp and it can be improved as follows when: and

Theorem 1.
Let the function f given by (1) belong to the class k-S * s (q). Then Proof. For f ∈ k-S * s (q), we have where and p k (z) is given by (5). The function p(z) with p(0) = 1 is given as follows: After some computation involving (16), we have Therefore, we find that We also have Comparing the corresponding coefficients in (17) and (18) along with Lemma 3, we obtain the required result.

Furthermore, we have
, , and Here P 1 and P 2 are given in (5).
Proof. By comparison of coefficients in (17) and (18), we can obtain A detailed calculation for T 3 (2) yields We need to maximize a 2 2 − 2a 2 3 + a 2 a 4 for f ∈ k-S * s (q). Thus, by writing a 2 , a 3 , a 4 in terms of c 1 , c 2 , c 3 , with the help of (19) and (21), we get Finally, applying the triangle inequality, Lemma 2 and Lemma 3 along with (22) and (23), we obtained the required result. Proof. Making use of (19), (20) and (21), we have where By using Lemma 1, we take Without loss of generality, we assume that c = c 1 (0 c 2) , so that Taking the moduli on both sides of (25) and using the triangle inequality, we find that This can be written as follows: where Hence, by putting Υ = 4 − c 2 1 and after some simplification, we have We consider G (c) = 0, for the optimum value of G(c), which implies that c = 0. Thus, G(c) has a maximum value at c = 0. Hence, the maximum value of G(c) is given by which occurs at c = 0 or Hence, by putting (27) and after some simplification, we obtain the desired result.
For q → 1−, k = 0, and p 1 = 2 in Theorem 3, we have the following known result for the class S * s .

Corollary 1.
(see [36]) If an analytic function f ∈ A that belongs to the class S * , then a 2 a 4 − a 2 3 1.

The Fekete-Szegö Problem
Theorem 4. Let the function f ∈ A given by (1) belong to the class k-S * s (q). Then Proof. From (19) and (20), we have By applying the triangle inequality and Lemma 4, we obtain Theorem 4.
If we set k = 0 and q → 1− in Theorem 4, we thus obtain the following known result.

Theorem 5.
If the function f given by (1) belongs to the class S * s (q) and if If we set k = 0 and q → 1−, we obtain the following known result.

Applications of the Main Results
In this section, firstly we recall that the Bernardi integral operator F β given in [38] as follows: The q-integral of the function f on [0, z] is defined as follows (see, for example [39]): and q-integral of the function z n is given by where n = −1 and for q → 1−, Equation (29) becomes Noor [39] introduced the q-Bernardi integral operator B q (z) as follows: Let f ∈ A. Then, by using Equations (29) and (8), we obtain the following power series for the function B q (z) in the open unit disk E as follows: [n + β] q a n z n .
Clearly, B q (z) is analytic in the open unit disk E.
Applying Theorem 1 on Equation (31), we obtain the following result.
Theorem 6. If the function B q (z) given by (31) belongs to the class k-S * s (q), where k ∈ [0, 1], then where B 2 , B 3 and B 4 are given in (32).
Applying Theorem 2 to Equation (31), we obtain the following result.
Applying Theorem 3 to Equation (31), we obtain the following result.
Theorem 8. If the function B q (z) given by (31) belongs to the class k-S * s (q), then For q → 1−, k = 0, β = 0 and p 1 = 2 in Theorem (8), we have the following known result for the class S * s .

Theorem 9.
If the function B q (z) given by (31) belongs to the class k-S * s (q), then If we set k = 0, β = 0 and q → 1− in Theorem 9, we obtain the following known result.

Conclusions
We have made use of the general conic domain Ω k,q and the quantum (or q-) calculus to introduce and investigate several new subclasses of q-starlike functions with respect to symmetrical points in open unit disk E. We have studied some interesting results such as the Hankel determinant, the Toeplitz matrices, and the Fekete-Szegö inequalities. We have also discussed some applications of our main results by using a q-Bernardi integral operator.
For further investigation, we can easily follow a known relationship between the q-analysis and (p, q)-analysis (see [25] (p. 340, Equations (9.1), (9.2) and (9.3))) and the results for the q-analogues, which we have included in this paper for 0 < q < 1, can then be easily transformed into the related results for the (p, q)-analogues with 0 < q < p 1 by adding a rather redundant (or superfluous) parameter p (see, for details [25] (p. 340)).

Conflicts of Interest:
The authors declare no conflict of interest.