An Upper Bound of the Third Hankel Determinant for a Subclass of q -Starlike Functions Associated with k -Fibonacci Numbers

: In this paper, we use q -derivative operator to deﬁne a new class of q -starlike functions associated with k -Fibonacci numbers. This newly deﬁned class is a subclass of class A of normalized analytic functions, where class A is invariant (or symmetric) under rotations. For this function class we obtain an upper bound of the third Hankel determinant.


Introduction and Definitions
The calculus without the notion of limits is called quantum calculus; it is usually called q-calculus or q-analysis. By applying q-calculus, univalent functions theory can be extended. Moreover, the q-derivative, such as the q-calculus operators (or the q-difference) operator, are used to developed a number of subclasses of analytic functions (see, for details, the survey-cum-expository review article by Srivastava [1]; see also a recent article [2] which appeared in this journal, Symmetry).
Ismail et al. [3] instigated the generalization of starlike functions by defining the class of q-starilke functions. A firm footing of the usage of the q-calculus in the context of Geometric Functions Theory was actually provided and the basic (or q-) hypergeometric functions were first used in Geometric Function Theory by Srivastava (see, for details [4]). Raghavendar and Swaminathan [5] studied certain basic concepts of close-to-convex functions. Janteng et al. [6] published a paper in which the (q) generalization of some subclasses of analytic functions have studied. Further, q-hypergeometric functions, the q-operators were studied in many recent works (see, for example, [7][8][9]). The q-calculus applications in operator theory could be found in [4,10]. The coefficient inequality for q-starlike and q-close-to-convex functions with respect to Janowski functions were studied by Srivastava et al. [8,11] recently, (see also [12]). Further development on this subject could be seen in [7,9,13,14]. For a comprehensive review of the theory and applications of the q-derivative (or the q-difference) operator and related literature, we refer the reader to the above-mentioned work [1].
We denote by A the class of functions which are analytic and having the form: in the open unit disk U given by U = {z : z ∈ C and |z| < 1} and normalized by the following conditions: The subordinate between two functions f and g in U, given by: if an analytic Schwarz function w exists in such way that In particular, the following equivalence also holds for the univalent function g Next by the P class of analytic functions, p(z) in U is denoted, in which normalization conditions are given as follow: such that (p (z)) > 0 (∀ z ∈ U) .
Let k be any positive real number, then we define the k-Fibonacci number sequence {F k,n } ∞ n=0 recursively by F k,0 = 0, F k,1 = 1 and F k,n+1 = kF k,n + F k,n−1 for n 1.
The n th k-Fibonacci number is given by Ifp k,n z n , then we have (see also [15]) p k,n = (F k,n−1 + F k,n+1 )T n k (n ∈ N; N := {1, 2, 3, · · · }). (5) Definition 1. Let q ∈ (0, 1) then the q-number [λ] q is given by Definition 2. The q-difference (or the q-derivative) D q operator of any given function f is defined, in a given subset of C, of complex numbers by led to the existence of the derivative f (0).
From Definitions 1 and 2, we have for a differentiable function f . In addition, from (1) and (2), we observe that [n] q a n z n−1 .
In the year 1976, it was Noonan and Thomas [16] who concentrated on the function f given in (1) and gave the qth Hankel determinant as follows.
Let n 0 and q ∈ N. Than the qth Hankel determinant is defined by Several authors studied the determinant H q (n). In particular, sharp upper bounds on H 2 (2) were obtained in such earlier works as, for example, in [17,18] for various subclasses of the normalized analytic function class A. It is well-known for the Fekete-Szegö functional a 3 − a 2 2 that Its worth mentioning that, for a parameter µ which is real or complex, the generalization the functional a 3 − µa 2 2 is given in aspects. In particular, Babalola [19] studied the Hankel determinant H 3 (1) for some subclasses of A.
In 2017, Güney et al. [20] explored the third Hankel determinant in some subclasses of A connected with the above-defined k-Fibonacci numbers. A derivation of the sharp coefficient bound for the third Hankel determinant and the conjecture for the sharp upper bound of the second Hankel determinant is also derived by them, which is employed to solve the related problems to the third Hankel determinant and to present an upper bound for this determinant.
Motivated and inspired by the above-mentioned work and also by the recent works of Güney et al. [20] and Uçar [12], we will now define a new subclass SL(k, q) of starlike functions associated with the k-Fibonacci numbers. We will then find the Hankel determinant H 3 (1) for the newly-defined functions class SL(k, q). Definition 3. Let P (β) (0 β < 1) denote the class of analytic functions p in U with p(0) = 1 and p (z) > β.

Definition 4.
Let the function p be said to belong to the class k-P q (z) and let k be any positive real number if wherep k (z) is given byp and T k is given in (4).

Remark 1.
For q = 1, it is easily seen that wherep k (z) is given in (8).
We recall that when the f belongs to the class A of analytic function then it is invariant (or symmetric) under rotations if and only if the function f ς (z) given by It can be easily checked that the functionals |a 2 a 3 − a 4 |, |H 2,1 | and |H 3,1 | considered for the class SL(k, q) satisfy the above definitions.
Lemma 1 (see [21]). Let be in the class P of functions with positive real part in U. Then Conversely, if p(z) ∼ = p 1 (z) for some |x| = 1, then c 1 = 2x and Lemma 2 (see [22]). Let p ∈ P with its coefficients c k as in Lemma 1, then Lemma 3 (see [23]). Let p ∈ P with its coefficients c k as in Lemma 1, then Lemma 4 (see [20]). If the function f given in the form (1) belongs to class SL k , then where T k is given in (4). Equality holds true in (14) for the function g given by which can be written as follows:

Main Results
Here, we investigate the sharp bounds for the second Hankel determinant and the third Hankel determinant. We also find sharp bounds for the Fekete-Szegö functional a 3 − λa 2 2 for a real number λ. Throughout our discussion, we will assume that q ∈ (0, 1).

Theorem 1.
Let the function f ∈ A given in (1) belong to the class SL(k, q). Then and T k is given in (4).
Proof. If f ∈ SL(k, q), then it follows from the definition that .
For a given f ∈ SL(k, q), we find for the function p(z), where then there is an analytic function w such that Therefore, the function g(z), given by is in the class P. It follows that From (5), we find the coefficientp k,n of the functionq given bỹ p k,n = (F k,n−1 + F k,n+1 )T n k .
This shows the following relevant connectionq with the sequence of k-Fibonacci numbers: If then, by (21) and (22), we find that Moreover, we have This can be written as follows: It is known that Applying (27) together with (11)-(13), we get From (27), we obtain which, for sufficiently large n, yields |c 1 | =: y ∈ [0, 2] .
After some computations, we can find that max y∈[0,2] x k,n − |B q |k 5 y 4 x k,n − |C q |k 3 y 4 x k,n + |B q |k 4 y 4 + |C q |k 2 y 4 As a result of the following limit formula: and by using (27) x k,n − |B q |k 5 y 4 x k,n − |C q |k 3 y 4 x k,n +|B q |k 4 y 4 + |C q |k 2 y 4 We thus find that If, in (20), we set then, by putting c 1 = c 2 = c 3 = 2 in (26), we obtain This completes the proof of Theorem 1.

Remark 3.
In the next result, for simplicity, we take the values of S q , L q and M q as given by

Theorem 2.
Let the function f ∈ A given in (1) belong to the class SL(k, q). Then Proof. Let f ∈ SL(k, q) and let p ∈ P be given in (2). Then, from (23)- (25) and we have which, together with (27), yields Now, applying the triangle inequality in (10)-(13), we get In addition, by using (27), we have and q 3 4 (q + 1) 2 (3k + 2) kx k,n − 3k 2 > 0 for 0 < k 1 and sufficiently large n. Therefore, we have got a function of the variable |c 1 | =: y ∈ [0, 2] and, after some computations, we can find that As a result of the following limit relation: and, by means of (27) If, in the formula (20) , we set then, by putting c 1 = c 2 = c 3 = 2 in (26), we obtain This completes the proof of Theorem 2.
This completes the proof of Theorem 4.

Conclusions
A new subclass of analytic functions associated with k-Fibonacci numbers has been introduced by means of quantum (or q-) calculus. Upper bound of the third Hankel determinant has been derived for this functions class. We have stated and proved our main results as Theorems 1-4 in this article.
Further developments based upon the the q-calculus can be motivated by several recent works which are reported in (for example) [24,25], which dealt essentially with the second and the third Hankel determinants, as well as [26][27][28][29], which studied many different aspects of the Fekete-Szegö problem.