Certain Class of Analytic Functions with respect to Symmetric Points Defined by Q -Calculus

In this study, we familiarise a novel class of Janowski-type star-like functions of complex order with regard to ( j, k ) -symmetric points based on quantum calculus by subordinating with pedal-shaped regions. We found integral representation theorem and conditions for starlikeness. Furthermore, with regard to ( j,k ) -symmetric points, we successfully obtained the coeﬃcient bounds for functions in the newly speciﬁed class. We also quantiﬁed few applications as special cases which are new (or known).


Definitions and Preliminaries
e set of all analytic functions constructed on the unit disc U � z ∈ C: |z| < 1 { } is symbolised by H(U). Also, A indicates the subclass of H(U) that has a Taylor series representation: e family of functions f ∈ A that are univalent in U is represented by S. is is well established that if f(ξ), assume by (1), is in S, then [f(ξ k )] 1/k (k is a positive integer) is consequently in S.
Definition 1 (see [1], Definition 3). Assume k is a positive integer. A domain D is known to be k-fold symmetric if a rotation of D about the origin through an angle 2π/k carries D onto itself. For U, a function f is said to be k-fold symmetric if and only if for each ξ in U f e 2πi/k ξ � e 2πi/k f(ξ). (2) F k represents the family including all k-fold symmetric functions. e concept of k-symmetrical function was protracted to so-called (j, k)-symmetrical function by Liczberski and Połubiński in [2]. To be specific, a function f(ξ) is reported for being (j, k)-symmetrical if where k ≥ 2 is a fixed integer, j � 0, 1, 2, . . . , k − 1 and ε � exp(2πi/k). e family of (j, k)-symmetrical functions will indeed be indicated by F j k . We believe that F 1 2 , F 0 2 , and F 1 k are quite well groups of odd functions, even functions, and k-symmetrical functions. Consider the subsequent equivalence demarcate f j,k (ξ) as well It is evident that f j,k (ξ) is a linear operator from U into U. If ] is an integer, then the subsequent assumptions result directly from (4): Let the function f ∈ A provided by (1) and g ∈ A of the form g(ξ) � ξ + ∞ n�2 Υ n ξ n , the Hadamard product (or convolution) of these two functions is indicated by Using Hadamard product, various authors studied the univalent function theory in dual with the theory of special functions, see [3][4][5] and references provided therein. roughout this whole article, we will assume that k ∈ N, ε � exp(2πi/k), and where f, g ∈ A; k � 1, 2, . . . ; j � 0, 1, 2, . . . , (k − 1).
From (7), we, thus, have e investigation of q-calculus (q stands for quantum) fascinated and inspired many scholars due its use in various areas of the quantitative sciences. Jackson [6,7] was among the key contributors of all the scientists who introduced and developed the q-calculus theory. Just like q-calculus was used in other mathematical sciences, the formulations of this idea are commonly used to examine the existence of various structures of function theory. ough it is the first article in which a link was established between certain geometric nature of the analytic function and the q-derivative operator and the usage of q-calculus in function theory, a solid and comprehensive foundation is given in [8] by Srivastava. After this development, many researchers introduced and studied some useful operators in q-analog with the applications of convolution concepts. For example, Kanas and Raducanu [9] established the q-differential operator and then examined the behavior of this operator in function theory. For more applications of this operator, see [10,11].
For f ∈ A assumed by (1) and 0 < q < 1, the Jackson's q-derivative operator or q-difference operator for f ∈ A is specified under (see [12][13][14]) From (10), if f is assumed as in (1), we can effortlessly see that [n] q a n ξ n− 1 , for ξ ≠ 0, provided the q-integer number [n] q is represented by and take into consideration lim . During our study, we let signify e q-Jackson integral is defined by (see [6]) If the q-series converges, further witness that where the second equality grasps if f is continuous at ξ � 0. Let the classes of star-like functions of order η (0 ≤ η < 1) and convex functions of order η (0 ≤ η < 1) are symbolised by S * (η) and C(η), respectively. In A, we categorize the collection P of functions p(ξ) ∈ A with p(0) � 1 and Rp(ξ) > 0. e functions in the P class are not univalent.
With U, let f, g be analytic. e function f is said to be subordinate to g in U if the Schwarz function ω(ξ) exists in U such that |ω(ξ)| < |ξ| and f(ξ) � g(ω(ξ)), as shown through f≺g. Whenever g is univalent in U, consequently the subordination is identical to f(0) � g(0) and f(U) ⊂ g(U).
Using the concept of subordination for holomorphic functions, Ma and Minda [15] proposed the classes: where ψ ∈ P with ψ ′ (0) > 0 maps U onto a region star-like with respect to 1 and symmetric with respect to the real axis. By making a choice ψ to map unit disc on to some specific regions such as cardioid, parabolas, lemniscate of Bernoulli, and booth lemniscate in the right-half of the complex plane, various interesting subclasses of star-like and convex functions could be gained well.
Lots of fascinating subclasses of star-like and convex functions may be constructed by using ψ to map unit disc on to particular areas such as cardioid, parabolas, lemniscate of Bernoulli, and booth lemniscate on the right-half of the complex plane.
For arbitrary fixed numbers C, D, − 1 < C ≤ 1, and − 1 ≤ D < C, we express through P(C, D) the family of functions p(ξ) � 1 + p 1 ξ + p 2 ξ 2 + · · · analytic in the unit disc and p(ξ) ∈ P(C, D) if and only if where w(ξ) is the Schwarz function. Geometrically, For detailed study on the class of Janowski functions, we refer [16]. e class of Janowski star-like functions and Janowski convex functions is defined as follows: Inspired by the theory familiarized by Sakaguchi [17], and the study on analytic functions with respect to (j, k)-symmetrical points by various authors (see [18][19][20][21][22]), under this article, we formulate new subclasses listed in Definition 2.
{ }, and H j,k (ξ)/ξ ≠ 0 be defined as in (7). We say that satisfies the subordination condition: where ψ ∈ P and is given by Remark 1. Here, we list few exceptional cases of the defined class K b s (ϑ; θ; ψ; g; C, D).
For completeness, we will now define q-analogue of the as follows. (7). We say that holds the subordination condition: where ψ ∈ P and ψ is defined as in (21).

Remark 2.
e impact of Janowski functions on a particular conic region was initiated by Noor and Malik [23] and was subsequently studied by various authors (see [11,24,25] and references provided therein).

Inclusion Relationships and Integral
Representations of the Classes K b s (ϑ; θ; ψ; g; C, D) and QK b s (ϑ; θ; ψ; g; C, D) Let us begin with the following.
Upon integration, we get or equivalently, is concludes the proof of eorem 3.

Journal of Mathematics
where I q f is the Jackson q-integral, defined as in (14). Integrating the above equality, we get or equivalently, is concludes the proof of eorem 3. By fixing C � 1, D � − 1, θ � 0, b � 1, and g(ξ) � ξ + ∞ n�2 ξ n in eorem 3, we state the subsequent result.

Coefficient Inequalities for
K b s (ϑ; θ; ψ; g; C, D) and QK b s (ϑ; θ; ψ; g; C, D) e coefficient estimate |a n | of the defined function classes is determined in this section.

Conclusion
Very few studies have been showed on analytic functions with regard to (j, k)-symmetric points. Since we have articulated the problem differently so as to deviate from the similar studies, only few special cases could be discussed. Furthermore, by swapping the ordinary differentiation with quantum differentiation, we have tried at the discretization of some of the familiar findings.

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.