Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions

The main purpose of this article is to find the upper bound of the third Hankel determinant for a family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator. The work is motivated by several special cases and consequences of our main results, which are pointed out herein.


Introduction
We denote by A (U) the class of functions which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1} , where C is the complex plane.Let A be the class of analytic functions having the following normalized form: in the open unit disk U, centered at the origin and normalized by the conditions given by f (0) = 0 and f (0) = 1.
In addition, let S ⊂ A be the class of functions which are univalent in U.The class of starlike functions in U will be denoted by S * , which consists of normalized functions f ∈ A that satisfy the following inequality: If two functions f and g are analytic in U, we say that the function f is subordinate to g and write in the form: f ≺ g or f (z) ≺ g (z) , if there exists a Schwarz function w which is analytic in U, with w (0) = 0 and |w (z)| < 1, such that f (z) = g w (z) .
Moreover, for two analytic functions f and g given by and the convolution (or the Hadamard product) of f and g is defined as follows: We next denote by P the class of analytic functions p which are normalized by such that p (z) > 0 (z ∈ U).
We now recall some essential definitions and concept details of the basic or quantum (q-) calculus, which are used in this paper.We suppose throughout the paper that 0 < q < 1 and that Definition 1.Let q ∈ (0, 1) and define the q-number [λ] q by Definition 2. Let q ∈ (0, 1) and define the q-factorial [n] q ! by Definition 3. Let q ∈ (0, 1) and define the generalized q-Pochhammer symbol [λ] q,n by Definition 4. For ω > 0, let the q-gamma function Γ q (ω) be defined by Definition 5. (see [3,4]) The q-derivative (or the q-difference) operator D q of a function f in a given subset of C is defined by provided that f (0) exists.
We note from Definition 5 that lim for a differentiable function f in a given subset of C. It is readily deduced from (1) and (4) that The operator D q plays a vital role in the investigation and study of numerous subclasses of the class of analytic functions of the form given in Definition 5. A q-extension of the class of starlike functions was first introduced in [5] by using the q-derivative operator (see Definition 6 below).A background of the usage of the q-calculus in the context of Geometric Funciton Theory was actually provided and the basic (or q-) hypergeometric functions were first used in Geometric Function Theory by Srivastava (see, for details, [6]).Some recent investigations associated with the q-derivative operator D q in analytic function theory can be found in [7][8][9][10][11][12][13] and the references cited therein.Definition 6. (see [5]) The notation S * q was first used by Sahoo et al. (see [14]).
It is readily observed that, as q → 1−, the closed disk given w − 1 1 − q 1 1 − q becomes the right-half plane and the class S * q reduces to S * .Equivalently, by using the principle of subordination between analytic functions, we can rewrite the conditions in ( 6) and (7) as follows (see [15]): (see [16]) For a function f ∈ A (U) , the Ruscheweyh-type q-derivative operator is defined as follows: where and From ( 8) it can be seen that and lim This shows that, in case of q → 1−, the Ruscheweyh-type q-derivative operator reduces to the Ruscheweyh derivative operator D δ f (z) (see [17]).From (8) the following identity can easily be derived: Now, by using the Ruscheweyh-type q-derivative operator, we define the following class of q-starlike functions.Definition 8.For f ∈ A (U) , we say that f belongs to the class RS * q (δ) if the following inequality holds true: or, equivalently, we have (see [15]) by using the principle of subordination.
Let n 0 and j 1.The jth Hankel determinant is defined as follows: The above Hankel determinant has been studied by several authors.In particular, sharp upper bounds on H 2 (2) were obtained by several authors (see, for example, [18-21]) for various classes of normalized analytic functions.It is well-known that the Fekete-Szegö functional a 3 − a 2 2 = H 2 (1).This functional is further generalized as a 3 − µa 2 2 for some real or complex µ.In fact, Fekete and Szegö gave sharp estimates of a 3 − µa 2 2 for real µ and f ∈ S, the class of normalized univalent functions in U.It is also known that the functional a 2 a 4 − a 2 3 is equivalent to H 2 (2).Babalola [22] studied the Hankel determinant H 3 (1) for some subclasses of analytic functions.In the present investigation, our focus is on the Hankel determinant H 3 (1) for the above-defined function class RS * q (δ) .

A Set of Lemmas
Each of the following lemmas will be needed in our present investigation.
Lemma 1. (see [23]) be in the class P of functions with positive real part in U.Then, for any complex number υ, When υ < 0 or υ > 1, the equality holds true in (13) if and only if or one of its rotations.If 0 < υ < 1, then the equality holds true in (13) if and only if or one of its rotations.If υ = 0, the equality holds true in (13) if and only if or one of its rotations.If υ = 1, then the equality in (13) holds true if p(z) is a reciprocal of one of the functions such that the equality holds true in the case when υ = 0.
Lemma 2. (see [24,25]) be in the class P of functions with positive real part in U. Then for some x, |x| 1 and for some z (|z| 1).

Main Results
In this section, we will prove our main results.Throughout our discussion, we assume that q ∈ (0, 1) and δ > −1.
Our first main result is stated as follows.
It is also asserted that, for and that , for Proof.If f ∈ RS * q (δ), then it follows from (12) that where We define a function p(z) by It is clear that p ∈ P. From the above equation, we have From (14), we find that Similarly, we get Therefore, we have and We thus obtain Finally, by applying Lemma 1 and Equation ( 13) in conjunction with (18), we obtain the result asserted by Theorem 1.
We now state and prove Theorem 2 below.
Theorem 2. Let f ∈ RS * q (δ) be of the form (1). Then Proof.From ( 15)-( 17), we obtain By using Lemma 2, we have Now, taking the moduli and replacing |x| by ρ and p 1 by p, we have where and Upon differentiating both sides (19) with respect to ρ, we have It is clear that ∂F(p, ρ) ∂ρ > 0, which show that F(p, ρ) is an increasing function of ρ on the closed interval [0, 1] .This implies that the maximum value occurs at ρ = 1.This implies that max{F(p, ρ)} = F(p, 1) =: G(p).
For p = 0, this shows that the maximum value of (G(p)) occurs at p = 0. Hence, we obtain The proof of Theorem 2 is thus completed.
If, in Theorem 2, we let q −→ 1− and put δ = 1, then we are led to the following known result.
Corollary 1. (see [18]) Let f ∈ S * .Then and the inequality is sharp. where Proof.Using the values given in (15) and (16) we have We now use Lemma 2 and assume that p 1 2. In addition, by Lemma 3, we let p 1 = p and assume without restriction that p ∈ [0, 2] .Then, by taking the moduli and applying the trigonometric inequality on (22) with ρ = |x| , we obtain where η (q) = q + q 2 + q 3 ψ 3 + 2q 3 − q 2 − 2q ψ 1 ψ 2 and κ (q) is given by (21).Differentiating F(ρ) with respect to ρ, we have This implies that F(ρ) is an increasing function of ρ on the closed interval [0, 1].Hence, we have that is, Since p ∈ [0, 2] , p = 2 is a point of maximum.We thus obtain which corresponds to ρ = 1 and p = 2 and it is the desired upper bound.
For δ = 1 and q → 1−, we obtain the following special case of Theorem 3.

Conclusions
By making use of the basic or quantum (q-) calculus, we have introduced a Ruscheweyh-type q-derivative operator.This Ruscheweyh-type q-derivative operator is then applied to define a certain subclass of q-starlike functions in the open unit disk U. We have successfully derived the upper bound of the third Hankel determinant for this family of q-starlike functions which are associated with the Ruscheweyh-type q-derivative operator.Our main results are stated and proved as Theorems 1-4.These general results are motivated essentially by their several special cases and consequences, some of which are pointed out in this presentation.