Applications of Symmetric Conic Domains to a Subclass of q -Starlike Functions

: In this paper, the theory of symmetric q -calculus and conic regions are used to deﬁne a new subclass of q -starlike functions involving a certain conic domain. By means of this newly deﬁned domain, a new subclass of normalized analytic functions in the open unit disk E is given. Certain properties of this subclass, such as its structural formula, necessary and sufﬁcient conditions, coefﬁcient estimates, Fekete–Szegö problem, distortion inequalities, closure theorem and subordination results, are investigated. Some new and known consequences of our main results as corollaries are also highlighted.


Introduction and Definitions
Let A denote the class of all analytic functions f which are analytic in the open unit disk E = {z : z ∈ C and |z| < 1} and series expansion is The usage of normalize functions in geometric function theory is significant. They are used to map everything inside a unit circle, where everything is convergent. Using this concept, many different subclasses of analytic functions have been defined and studied.
Furthermore, let S ⊂ A represents the set of all univalent in E (see [1,2]). The classes of uniformly convex (U CV ) and uniformly starlike (U ST ) functions were introduced by Goodman [3] and are defined analytically as: The convolution of two analytic functions f (z) = ∞ ∑ n=0 a n z n and g(z) = ∞ ∑ n=0 b n z n , (z ∈ E) is defined as: ( f * g)(z) = ∞ ∑ n=0 a n b n z n .
Let P represent the set of all of Carathéodory functions and every analytic function p ∈ P, having series of the form p(z) = 1 + ∞ ∑ n=1 c n z n and satisfying the condition (p(z)) > 0, z ∈ E.
In terms of these analytic functions, many different subclasses of starlike and convex functions have been defined already.
We have discussed above that Kanas and Wisniowska [4] introduced the class of k-uniformly convex functions (k-U CV ) and the corresponding class of k-starlike functions (k-ST ) and then defined these classes subject to the conic domain Ω k , (k ≥ 0) as follows: or Ω k = {w : w > k|w − 1|}.
For k = 0, then (2) becomes the right half plane; for 0 < k < 1, it becomes a hyperbola, for k = 1 it becomes a parabola and for k > 1, it becomes an ellipse. The generalization of theses discussed by Shams et al. in [7] is subject to the conic domain Ω k,γ (k ≥ 0), γ ∈ C\{0}, which is Ω k,γ = u + iv : u > k (u − 1) 2 or For this conic domain, the function p k,γ (z) plays the role of the extremal function for the conic domain Ω k,γ .
and i ∈ (0, 1), k = cosh πK (i) 4K(i) , K(i) is the first kind of Legendre's complete elliptic integral. For details (see [4]). Indeed, from (3), we have where and In the twentieth century the researchers have developed a great interest in the study of theory of q-calculus and its numerous applications in the fields of mathematics and physics. In defining the q-analogous of the derivative and integral operator and providing few of their applications, Jackson [8] was one of the pioneer researchers. It has several applications in number theory, combinatorics, orthogonal polynomials, fundamental hyper-geometric functions, and other fields of mathematics, including quantum mechanics, mechanics and relativity theory. Furthermore, quantum calculus has been proven to be a branch of the more comprehensive mathematical area of time scale calculus. For both discrete and continuous domains, time scales provide a coherent framework for studying dynamic equations. Many researchers provided useful applications for q-analysis in domains of mathematics; see [9][10][11][12][13][14][15][16][17][18]. Currently, operators of basic (or q-) calculus and fractional qcalculus and their applications have been elaborated by Srivastava in [19]. Considering the importance of q-operator calculus theory, researchers have intensively explored their applications in various fields; see [20][21][22][23][24][25].
In many areas of mathematics, the symmetric properties of functions play an important role in problem solving. Symmetry's importance has been demonstrated in a variety of fields, including biology, chemistry, and psychology. Recently, in [26], Zhang et al. studied the application of a q-symmetric difference operator in harmonic univalent functions. Additionally, Khan et al. [27] defined the symmetric conic domain by using symmetric q-calculus operator theory and investigated q-starlike functions in this domain. Very recently, Khan et al. [28] defined and explored a new subclass of analytic and bi-univalent function involving certain q-Chebyshev polynomials by means of a symmetric q-operator. The symmetric q-calculus finds its applications in different fields, especially in quantum mechanics; see for example [29,30].
Here we present a few basic definitions, as well as a concept of q-calculus and symmetric q-calculus which will help us in further studies.

Remark 1.
The symmetric q-number can not reduce to a q-number.

Definition 5 ([32]
). Let f ∈ A; then, the symmetric q-derivative operator is defined by: We can write (8) as: [n] q a n z n−1 .
Note that [n] q a n z n−1 ,
By taking motivation from the above-cited works, we define the following domain: (11) and p k,γ (z) is given by (3). Geometrically, the function p(z) ∈ k-P q,q −1 ,γ (z) takes on all values from the domain Ω k,q,q −1 ,γ which is defined as: where where Ω k,γ is the conic domain considered by Shams et al. [7]. Secondly, we have where Ω k is the conic domain considered by Kanas and Wisniowska [34]. Thirdly, we have where P (p k ) is the well-known class introduced by Kanas and Wisniowska [34]. Lastly, we have where P is the well-known class of analytic functions with a positive real part.
Using the above-defined domain, we now define the following subclass of certain analytic functions: where Remark 3. First of all, we have (see [34]) Secondly, it could be seen that (see [4]) Geometrically, the function f ∈ A is in the class k-U ST (q, q −1 , γ), if and only if the function J (q, q −1 , f (z)) takes all values in the conic domain Ω k,q,q −1 ,γ given by (12). Taking this geometrical interpretation into consideration, one can rephrase the above definition as: where p k,q,q −1 ,γ (z) is defined by (11).
We also fixed k- Our further investigation is organized as follows. In Section 2, we give some supporting results in form of Lemmas, which will help in order to obtain our new results in Section 3. In Section 3, we obtain our main results and also give some of their special cases in the form of Corollaries and Remarks. In Section 4, we conclude our present investigation and also give some future direction to the interested readers toward the prospect that this kind of result will be obtained for other new subclasses of analytic functions.

A Set of Lemmas
We need the following lemmas in order to prove our main results.
Proof. From (11), we have By using (4) in (18), we have The series are convergent and convergent to 1, Therefore, (19) becomes This complete the proof of Lemma 2.
By using (20) in (21), we have Now by using Lemma 1 on (22), we have Hence, the proof of Lemma 3 is complete.

Main Results
We now state and prove our main results. The first theorem of this section gives necessary conditions for an analytic function f of the form (1) to belong to the newly defined class k-U ST (q, q −1 , γ).

Theorem 1. If an analytic function f is of the form (1), and it satisfies
Proof. Suppose that (23) holds; then, it is sufficient to show that Using (15), we have The expression (41) is bounded above by 1.
After some simple calculations, we have Hence, we complete the proof of Theorem 1.

Corollary 1. An analytic function f of the form (1) belongs to the class k-U ST (α) if it satisfies the condition
where 0 ≤ α < 1 and k ≥ 0.
Theorem 2. If f ∈ k-U ST (q, q −1 , γ) and is of the form (1), then and where Q 1 and ϕ j are defined by (5) and (29).
Equating coefficients of z n on both sides, we have This implies that By using Lemma 3, we have where Now we prove that For this, we use the induction method. For n = 2 from (28), we have From (26), we have For n = 3, from (28), we have From (26), we have Let the hypothesis be true for n = m. From (28), we have From (26), we have By the induction hypothesis, we have Multiplying That is which shows that inequality (31) is true for n = m + 1. Hence, the proof of Theorem 2 is now completed.
In the next Theorem, we state and prove the Fekete-Szegö-type result for our defined function class k-U ST (q, q −1 , γ). Theorem 3. Let 0 ≤ k < ∞, q ∈ (0, 1) be fixed, and let f (z) ∈ k-U ST (q, q −1 , γ) and be of the form (1). Then for a complex number µ, where v is given by (37).
Proof. If f (z) ∈ k-U ST (q, q −1 , γ), then we have J (q, q −1 , f (z)) ≺ p k,q,q −1 ,γ (z) and if there exists a Schwarz function w(z), we have Let the function h(z) ∈ P, defined by: this gives Now, from L.H.S of (33), we have By using (34) and (35) in (33), we obtain For any complex number µ we have where Now by using Lemma 4 on (36), we have Hence, we complete the proof of Theorem 3.
Next we investigate the necessary and sufficient conditions for f (z) of the form (17) to be in the class k-U ST − (q, q −1 , γ). Theorem 4. Let k ∈ [0, ∞), q ∈ (0, 1) and γ ∈ C\{0}. A function f (z) of the form (17) will belong to the class k-U ST − (q, q −1 , γ) which can be expressed as: The result is sharp for the function Proof. In view of Theorem 1, it remains to prove the necessity. If f (z) ∈ k-U ST − (q, q −1 , γ), then infect that | (z)| ≤ |z|; for any z, we have Letting z → 1− along the real axis, we obtain the desired inequality (38). Hence we complete the proof of Theorem 4.
Then f ∈ k-U ST − (q, q −1 , γ), if and only if f can be expressed in the form of λ n f n (z), λ n > 0, and where R(n, q) is given by (44).
Conversely, suppose that k-U ST − (q, q −1 , γ). Since |a n | ≤ R(n, q), we can set |a n |, and Then f (z) = z + ∞ ∑ n=2 a n z n The proof of Theorem 5 is complete.
By letting r → 1−, we get the required result. Hence, the proof of Theorem 6 is complete.
Hence, proof of Theorem 7 is complete.

Conclusions
In this paper, motivated significantly by a number of recent works, we used the concept of symmetric quantum calculus and conic regions to define a new domain Ω k,q,q −1 ,γ , which generalizes the symmetric conic domains. By using a certain generalized symmetric conic domain Ω k,q,q −1 ,γ , we defined and investigated a new subclass of normalized analytic q-starlike functions in the open unit disk E, and we have successfully derived several properties and characteristics of a newly defined subclass of analytic functions. For the verification and validity of our main results, we have also pointed out relevant connections of our main results with those in several earlier related works on this subject. To conclude our present investigation, we would like to remark that one may attempt the results presented in this paper for different subclasses of analytic functions in different domains. In particular, one may define a new subclass of symmetric q-starlike functions associated with this newly defined domain and can obtain the same results.