Hankel and Toeplitz Determinants for a Subclass of q-Starlike Functions Associated with a General Conic Domain

By using a certain general conic domain as well as the quantum (or q-) calculus, here we define and investigate a new subclass of normalized analytic and starlike functions in the open unit disk U. In particular, we find the Hankel determinant and the Toeplitz matrices for this newly-defined class of analytic q-starlike functions. We also highlight some known consequences of our main results.


Introduction and Definitions
Let the class of functions, which are analytic in the open unit disk U = {z : z ∈ C and |z| < 1} , be denoted by L (U).Also let A denote the class of all functions f , which are analytic in the open unit disk U and normalized by f (0) = 0 and f (0) = 1.
Then, clearly, each f ∈ A has a Taylor-Maclaurin series representation as follows: Suppose that S is the subclass of the analytic function class A, which consists of all functions which are also univalent in U.
A function f ∈ A is said to be starlike in U if it satisfies the following inequality: We denote by S * the class of all such starlike functions in U.
For two functions f and g, analytic in U, we say that the function f is subordinate to the function g and write this subordination as follows: 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) .
In the case when the function g is univalent in U, then we have the following equivalence (see, for example, [1]; see also [2]): Next, for a function f ∈ A given by (1) and another function g ∈ A given by the convolution (or the Hadamard product) of f and g is defined here by Let P denote the well-known Carathéodory class of functions p, analytic in the open unit disk U, which are normalized by such that p (z) > 0 (z ∈ U) .
Following the works of Kanas et al. (see [3,4]; see also [5]), we introduce the conic domain Ω k (k 0) as follows: In fact, subjected to the conic domain Ω k (k 0), Kanas and Wiśniowska (see [3,4]; see also [6]) studied the corresponding class k-S T of k-starlike functions in U (see Definition 1 below).For fixed k, Ω k represents the conic region bounded successively by the imaginary axis (k = 0), by a parabola (k = 1), by the right branch of a hyperbola (0 < k < 1), and by an ellipse (k > 1).
For these conic regions, the following functions play the role of extremal functions. where and κ ∈ (0, 1) is so chosen that Here K(κ) is Legendre's complete elliptic integral of first kind and that is, K (κ) is the complementary integral of K (κ) (see, for example, ( [7], p. 326, Equation 9.4 (209))).Indeed, from (5), we have The class k-S T is defined as follows.
Definition 1.A function f ∈ A is said to be in the class k-S T if and only if We now recall some basic definitions and concept details of the q-calculus which will be used in this paper (see, for example, ( [7], p. 346 et seq.)).Throughout the paper, unless otherwise mentioned, we suppose that 0 < q < 1 and Definition 2. Let q ∈ (0, 1) and define the q-number [λ] q by Definition 3. Let q ∈ (0, 1) and define the q-factorial [n] q ! by Definition 4 (see [8,9]).The q-derivative (or q-difference) operator D q of a function f defined, in a given subset of C, by provided that f (0) exists.
From Definition 4, we can observe that lim for a differentiable function f in a given subset of C. It is also known from ( 1) and ( 7) that Definition 5.The q-Pochhammer symbol [ξ] n,q (ξ ∈ C; n ∈ N 0 ) is defined as follows: [ξ] n,q = q ξ ; q n Moreover, the q-gamma function is defined by the following recurrence relation: Definition 6 (see [10]).For f ∈ A, let the q-Ruscheweyh derivative operator R λ q be defined, in terms of the Hadamard product (or convolution) given by (2), as follows: We next define a certain q-integral operator by using the same technique as that used by Noor [11].
Definition 7.For f ∈ A, let the q-integral operator F q,λ be defined by where Clearly, we have We note also that, in the limit case when q → 1−, the q-integral operator F q,λ given by Definition 7 would reduce to the integral operator which was studied by Noor [11].
The following identity can be easily verified: When q → 1−, this last identity in (10) implies that which is the well-known recurrence relation for the above-mentioned integral operator which was studied by Noor [11].
In geometric function theory, several subclasses belonging to the class of normalized analytic functions class A have already been investigated in different aspects.The above-defined q-calculus gives valuable tools that have been extensively used in order to investigate several subclasses of A. Ismail et al. [12] were the first who used the q-derivative operator D q to study the q-calculus analogous of the class S * of starlike functions in U (see Definition 8 below).However, a firm footing of the q-calculus in the context of geometric function theory was presented mainly and basic (or q-) hypergeometric functions were first used in geometric function theory in a book chapter by Srivastava (see, for details, ( [13], p. 347 et seq.);see also [14]).Definition 8 (see [12]).A function f ∈ A is said to belong to the class S * q if It is readily observed that, as q → 1−, the closed disk: becomes the right-half plane and the class S * q of q-starlike functions reduces to the familiar class S * of normalized starlike functions in U with respect to the origin (z = 0).Equivalently, by using the principle of subordination between analytic functions, we can rewrite the conditions in (11) and (12) as follows (see [15]): The notation S * q was used by Sahoo and Sharma [16].Now, making use of the principle of subordination between analytic functions and the above-mentioned q-calculus, we present the following definition.Definition 9. A function p is said to be in the class k-P q if and only if , where p k (z) is defined by (5).
Geometrically, the function p (z) ∈ k-P q takes on all values from the domain Ω k,q (k 0) which is defined as follows: The domain Ω k,q represents a generalized conic region.
It can be seen that lim where Ω k is the conic domain considered by Kanas and Wiśniowska [3].Below, we give some basic facts about the class k-P q .
Remark 1.First of all, we see that where P 2k 2k+1+q is the well-known class of functions with real part greater than 2k 2k+1+q .Secondly, we have where P (p k ) is the well-known function class introduced by Kanas and Wiśniowska [3].Thirdly, we have lim q→1− 0-P q = P, where P is the well-known class of analytic functions with positive real part.
Definition 10.A function f is said to be in the class ST (k, λ, q) if and only if or, equivalently, Remark 2. First of all, it is easily seen that where S * q is the function class introduced and studied by Ismail et al. [12].Secondly, we have where k-S T is a function class introduced and studied by Kanas and Wiśniowska [4].Finally, we have where S * is the well-known class of starlike functions in U with respect to the origin (z = 0).
Remark 3. Further studies of the new q-starlike function class ST (k, λ, q) , as well as of its more consequences, can next be determined and investigated in future papers.
Let n ∈ N 0 and j ∈ N. The following jth Hankel determinant was considered by Noonan and Thomas [17]: , where a 1 = 1.In fact, this determinant has been studied by several authors, and sharp upper bounds on H 2 (2) were obtained by several authors (see [18-20]) for various classes of functions.It is well-known that the Fekete-Szegö functional a 3 − a 2 2 can be represented in terms of the Hankel determinant as H 2 (1).This functional has been further generalized as a 3 − µa 2  2 for some real or complex µ.Fekete and Szegö gave sharp estimates of a 3 − µa 2  2 for µ real 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) (see [18]).Babalola [21] studied the Hankel determinant H 3 (1) for some subclasses of normalized analytic functions in U.The symmetric Toeplitz determinant T j (n) is defined by . . .a n , so that and so on.For f ∈ S, the problem of finding the best possible bounds for ||a n+1 | − |a n || has a long history (see, for details, [22]).It is a known fact from [22] that for a constant c.However, the problem of finding exact values of the constant c for S and its various subclasses has proved to be difficult.In a very recent investigation, Thomas and Abdul-Halim [23] succeeded in obtaining some sharp estimates for T j (n) for the first few values of n and j involving symmetric Toeplitz determinants whose entries are the coefficients a n of starlike and close-toconvex functions.
In the present investigation, our focus is on the Hankel determinant and the Toeplitz matrices for the function class ST (k, λ, q) given by Definition 10.

A Set of Lemmas
In order to prove our main results in this paper, we need each of the following lemmas.
Lemma 1 (see [20]).If the function p (z) given by (3) is in the Carathéodory class P of analytic functions with positive real part in U, then for some x, z ∈ C with |x| 1 and |z| 1.
Lemma 2 (see [24]).Let the function p(z) given by (3) be in the Carathéodory class P of analytic functions with positive real part in U. Also let µ ∈ C. Then Lemma 3 (see [22]).Let the function p(z) given by (3) be in the Carathéodory class P of analytic functions with positive real part in U.
This last inequality is sharp.
Theorem 1.If the function f (z) given by (1) belongs to the class ST (k, λ, q) , where k ∈ [0, 1] , then where p j (j = 1, 2, 3) are positive and are the coefficients of the functions p k (z) defined by (6).Each of the above results is sharp for the function g (z) given by .
Proof.Let f (z) ∈ ST (k, λ, q).Then, we have where , and the functions p k (z) are defined by (6).We now define the function p (z) with p (0) = 1 and with a positive real part in U as follows: After some simple computation involving ( 16), we get We thus find that Now, upon expanding the left-hand side of (15), we have Finally, by comparing the corresponding coefficients in ( 17) and ( 18) along with Lemma 3, we obtain the result asserted by Theorem 1.
Theorem 2. If the function f (z) given by (1) belongs to the class ST (k, λ, q) , then where , , , , and p j (j = 1, 2) are positive and are the coefficients of the functions p k (z) defined by (6).
Proof.Upon comparing the corresponding coefficients in ( 17) and ( 18), we find that By a simple computation, T 3 (2) can be written as follows: We need to maximize a 2 2 − 2a 2 3 + a 2 a 4 for a function f ∈ ST (k, λ, q).So, by writing a 2 , a 3 , and a 4 in terms of c 1 , c 2 , and c 3 , with the help of ( 19)-( 21), we get Finally, by applying the trigonometric inequalities, Lemmas 2 and 3 along with (22), we obtain the result asserted by Theorem 2.
As an application of Theorem 2, we first set ψ n−1 = 1 and k = 0 and then let q → 1− .We thus arrive at the following known result.
Hence, by puting Y = 4 − c 2 1 and after some simplification, we have For optimum value of G (c), we consider G (c) = 0, which implies that c = 0.So G (c) has a maximum value at c = 0. We therefore conclude that the maximum value of G (c) is given by 1 4q 2 ψ 2 2 p 2 1 , which occurs at c = 0 or This completes the proof of Theorem 3.
If we put ψ n−1 = 1 and let q → 1− in Theorem 3, we have the following known result.

Concluding Remarks and Observations
Motivated significantly by a number of recent works, we have made use of a certain general conic domain and the quantum (or q-) calculus in order to define and investigate a new subclass of normalized analytic functions in the open unit disk U, which we have referred to as q-starlike functions.For this q-starlike function class, we have successfully derived several properties and characteristics.In particular, we have found the Hankel determinant and the Toeplitz matrices for this newly-defined class of q-starlike functions.We also highlight some known consequences of our main results which are stated and proved as theorems and corollaries.