Hankel and Symmetric Toeplitz Determinants for a New Subclass of q -Starlike Functions

: This paper considers the basic concepts of q -calculus and the principle of subordination. We deﬁne a new subclass of q -starlike functions related to the Salagean q -differential operator. For this class, we investigate initial coefﬁcient estimates, Hankel determinants, Toeplitz matrices


Introduction and Definitions
Let the set of all analytic functions g in the open unit disk E = {z : z ∈ C and |z| < 1} be denoted by A and every g ∈ A can be expressed as g(z) = z + ∞ ∑ n=2 a n z n . (1) Let S be the subset of A, whose functions are univalent in E. A function g ∈ A is known as a starlike function (denoted g ∈ S * ) and a convex function (denoted g ∈ K) if it satisfies the following inequalities.
In particular, if we take k = 0, then k − U S * = U S * and k − U CV = U CV introduced by Goodman [3]. Moreover, Wang et al. [4] defined and investigated the subclasses S * (α, β) and K(α, β) of analytic functions satisfy the following conditions, respectively.
The study of q-calculus has recently inspired researchers because of its many applications in mathematics and physics, especially in quantum physics. Jackson [5,6] was the first who introduced the q-analogues of derivatives by applying the q-calculus theory. He defined the q-derivative operator (D q ) for analytic function g in the open unit disk U. Furthermore, in [7], Ismail et al. defined q-starlike functions by using the quantum (or q-) calculus operator theory, and many researchers studied q-calculus in the perspective of Geometric Functions Theory (GFT). In 2014, Kanas and Raducanu [8] introduced Ruscheweyh q-differential operators and discussed some of its applications in a class of analytic functions related to conic domains. After that, many q-differential and integral operators have been defined so far (see for details [9,10]. Arif et al. [11,12] studied q-derivative operator for multivalent functions, and in [13] Zang et al. gave the generalizations of the conic domain by using q-calculus. Srivastava [14] used fractional q-calculus operators to discuss some properties of analytic functions. Recently, Srivastava [15] published a review article that benefits new researchers and scholars that are working in GFT and q-calculus. Khan et al. [16,17] studied the q-derivative operator and defined a new subclass of q-starlike functions, while in [18] Mahmood et al. investigated a third Hankel determinant for the class of q-starlike functions. Presently, we recall some definitions and details about q-calculus, which will help us to understand this new article. Definition 1 ([19]). The q-number [t] q for q ∈ (0, 1) is defined as In particular, t = n ∈ N, The q-factorial [n] q ! can be defined as [k] q , (n ∈ N).
Definition 3 (Jackson [6]). The q-integral for a function g is defined by g(q n z)q n .

Definition 4 ([5]
). For g ∈ A, the q-derivative operator or q-difference operator is defined by [n] q a n z n−1 .

Definition 5 ([20]
). The Salagean q-differential operator for g is defined by [n] m q a n z n .
Motivated by the work of Kanas and Raducanu [8] and Govindaraj and Sivasubramanian [20], we define the following class of functions with the help of q-calculus. Definition 6. An analytic function g is said to be in the class S * (m, q, α, β) if where Note that S * (m, q, α, β) ⊂ S * ⊂ S.
The Hankel determinant plays an important role in the theory of singularities [23] and are helpful in the study of power series with integer coefficients (see [24][25][26]). Note that the number of authors found the sharp upper bounds on H 2 (2) (see, for example, [27][28][29][30][31] for numerous classes of functions.
If j = 2 and n = 1, we then obtain a well-known fact for the Fekete-Szegö functional that: This functional was further generalized as follows, for some real or complex number µ. If j = 2 and n = 2, then Janteng [32] defined the following Hankel determinant and studied it for starlike functions.
The problem of determining the sharp upper bounds for the functional a 2 − µa 2 2 for a given compact family F of functions in the normalized analytic class A is often called the Fekete-Szegö problem for F . Many researchers have investigated the Fekete-Szegö problem for analytic functions (see [45][46][47]).
Aleman and Constantin [48] produced an admirable connection between univalent function theory and fluid dynamics. They found explicit solutions to the incompressible two-dimensional Euler equations by means of a univalent harmonic mapping. More accurately, the problem of finding all solutions describing the particle paths of the flow in Lagrangian variables was reduced to finding harmonic functions satisfying an explicit nonlinear differential system in C n with n = 3 or n = 4 (see also [49]). The problem of finding the best possible bounds for ||a n+1 | − |a n || has a long history (see [50]). For more details about the symmetric Toeplitz determinants, see [51,52].

A Set of Lemmas
In this section, we give some lemmas to investigate the main results of this paper.

Lemma 1 ([50]
). Let the function p(z) be given by (2), then The inequality is sharp for the following function, Lemma 2 ([53,54]). Let for some x, z ∈ C, with |z| ≤ 1 and |x| ≤ 1. Let the function p(z) be analytic in E and given by (2), then

Main Results
In the following theorem, we will find initial coefficients bounds, which will help out to prove other results. Theorem 1. Let the function g of the form (1) be in the class S * (m, q, α, β). Then where Proof. Let g ∈ S * (m, q, α, β), then we have where, After some simple calculations, we obtain Let then In view of (9), (10) and (12), we have ϕ(u(z)) Similarly, Equating the corresponding coefficients of (13) and (14), we have where Applying the Lemma 1 on (15)-(17), we obtained the desired result after some simplification.
We take q → 1−, m = 1, β = 1, and α = 1, we then have the following corollary proved in [44]. Corollary 1 ([44]). Let the function g of the form (1) be in the class S * . Then In Theorem 3, we will investigate the second Hankel determinant H 2 (2). Theorem 3. Let the function g of the form (1) be in the class S * (m, q, α, β). Then Proof. Making use of (15), (16), and (17), we obtain where By using Lemma 2 and taking Υ = 4 − c 2 1 and Z = 1 − |x| 2 z. Without loss of generality, we assume that c = c 1 , (0 ≤ c ≤ 2), so that where Applying for the modulus on both sides of (28) and using a triangle inequality, Putting Υ = 4 − c 2 1 , after some simplification, we have Let Q (c) = 0, the optimum value of Q(c) implies that c = 0. So Q(c) has the maximum value at c = 0, which is given by which occurs at c = 0 or  (29), we obtained the desired result.

Proof. From
Applying Lemma 2, if c = c 1 (0 ≤ c ≤ 2), then Applying the triangle inequality, we deduce It follows that where Therefore, where, So we can obtain the required result (30) by using Equations (35) and (36) in inequality (34).

Corollary 3 ([56]
). Let the function g of the form (1) be in the class S * . Then Theorem 5. Let the function g of the form (1) be in the class S * (m, q, α, β). Then where Proof. It follows from (15) and (16) that where, Λ 3 is given by (31). Now by using Lemma 3 on (39), we get the required result.

Applications
In this section, we provide q-analogue of the Bernardi integral operator to discuss some applications of our main results.
In [57], Noor et al. defined q-analogue of Bernardi integral operator for analytic functions g ∈ A as follows: [β + 1] q [n + β] q a n z n , z ∈ E, β > −1, B n a n z n .
Theorem 6. Let the function g of the form (1) be in the class S * (m, q, α, β) and B q β (z) is given by (41). Then Proof. The proof follows easily by using (41) and Theorem 1.

Theorem 7.
If the function B q β (z) is given by (41) belongs to the class S * (m, q, α, β). Then Proof. The proof follows easily by using (41) and Theorem 3.

Theorem 8.
If the function B q β (z) is given by (41) belongs to the class S * (m, q, α, β). Then Proof. The proof follows easily by using (41) and Theorem 5.

Conclusions
The work presented in this paper is motivated by the well-established usage of the basic (or q-) calculus in the context of Geometric Function Theory. For this class, we investigated Hankel determinants, Toeplitz matrices and Fekete-Szegö problems. Moreover, the q-Bernardi integral operator is used to discuss some applications of the main results of this paper. Moreover, for validity of our results, the relevant connections with those in earlier works are also pointed out.
In a review article [15], Srivastava explained that (p, q)-calculus was exposed to be a rather trivial and inconsequential variation of the classical q-calculus and the additional parameter p being redundant or superfluous (for detail see [37], p. 340). According to this observation of Srivastava [15] will indeed apply to any attempt to produce the rather straightforward and inconsequential (p, q)-variations of the results, which we have proved in this paper.