Geometric Properties of Certain Classes of Analytic Functions with Respect to ( x , y ) -Symmetric Points

: In this article, the present study employs the utilization of the concepts pertaining to ( x , y ) - symmetrical functions, Janowski type functions, and q -calculus in order to establish a novel subclass within the open unit disk. Speciﬁcally, we delve into the examination of convolution properties, which serve as a tool for investigating and inferring adequate and equivalent conditions. Moreover, we also explore speciﬁc characteristics of the class (cid:101) S x , y q ( α , β , λ ) , thereby further scrutinizing the convolution properties of these newly deﬁned classes


Introduction
The theory of functions exhibiting (x, y)-symmetry has a wide range of intriguing applications.For instance, these functions are useful in exploring the set of fixed points of mappings, estimating the absolute value of certain integrals, and deriving results akin to Cartan's uniqueness theorem for holomorphic mappings, as demonstrated in [1].The intrinsic properties of (x, y)-symmetrical functions are of great interest in the field of Geometric Function Theory.In this work, we present fundamental definitions and concepts related to (x, y)-symmetrical functions.We assume that y is a fixed integer and set ε = e 2πi y .Furthermore, we definedomain G ⊂ C to be a y-fold symmetric domain if G ⊂ C. Function h is referred to as a y-symmetrical function if, for every ζ ∈ G, h(εζ) = εh(ζ).
Let F (k) be the collection of all functions that possess analyticity within the open unit disk k = {ζ ∈ C : |ζ| < 1} and let F denote a particular subfamily, which is characterized by classh ∈ F (k), and takes the form of consider S as the set of all functions belonging to F that are univalent within k.The convolution or Hadamard product of two analytic functions h and g, both belonging to the F set, can be expressed as where h is defined by Equation ( 4) and Those interested in the Hadamard product and its generalization are encouraged to refer to sources [2,3].To introduce new classes of q-Janowski symmetrical functions defined in k, it is necessary to first review the relevant concepts and notations pertaining to Janowski-type functions, (x, y)symmetrical functions, and quantum calculus, also known as q-calculus.In his work [4], Janowski introduced the class P [α, β] which pertains to a given h ∈ F and h Here, ∆ represents the family of Schwarz functions, that is, In [5], Jackson presented and examined the notion of the q-derivative operator ∂ q h(ζ) where q satisfies the condition 0 < q < 1.The operator is defined as follows Alternatively, as expressed in (6), it can be written as Note that as Then, lim where h (ζ) is the ordinary derivative.
The q-integral of a function h, which was presented by Jackson [6], is regarded as a right inverse given the condition that provided that the series ∑ ∞ v=0 q v h(ζq v ) converges.Ismail et al. [7] introduced the initial link between quantum calculus and geometric function theory.In recent years, numerous authors have utilized the quantum calculus approach to investigate the geometric properties of several subclasses of analytic functions.For instance, Naeem et al. [8] delved into the subclass of q-convex functions, while Srivastava et al. [9] examined subclasses of q-starlike functions.Additionally, Alsarari and Alzahrani [10] explored the convolution conditions of q-Janowski symmetrical function classes.Moreover, Ovindaraj and Sivasubramanian in [11] discovered subclasses related to q-conic domains, and several recent studies can be found in [12][13][14].Khan et al. [15] employed the symmetric q-derivative operator, while Srivastava [16] published a survey-cum-expository review paper that is highly beneficial to researchers.

Proposition 1.
For n and m any real (or complex) constants and ζ ∈ k, we have 1. .
The function h is denoted as λ-spiral-like if the real part of e iλ ζh (ζ) is greater than zero.Here, λ represents a real number such that its absolute value is less than π 2 .In recent research conducted by Al sarari et al. [17,18], numerous intriguing findings were obtained for various classes by utilizing the concept of (x, y)-symmetrical functions and the q-derivative.Consequently, we combine the notion of (x, y)-symmetrical functions, the q-derivative, and Janowski-type functions to establish the ensuing classes.Definition 1.For arbitrary fixed numbers q, α, β and λ, 0 where h x,y is defined in (1).

•
If we substitute µ v = v in Definition 2 we obtain the concept initially introduced by Goodman [22] and later generalized by Ruscheweyh [23].
, we arrive at the definition of a neighborhood with q-derivative N µ q,ρ (h), N µ q,ρ (e), where [v] q is defined by Equation (7).

Main Results
In this section, we investigate the convolution conditions for the class of convex functions in Theorem 2, which will be used to get the convolution conditions for the class of star-like functions in Theorem 3, which is the equivalent of Remark 4. Theorem 5, with its application of Corollary 2, will be used as a supporting result to deduce the sufficient condition of Theorem 4 and Corollary 1, which will be used to get the neighborhood results of Theorems 6 and 7.
where u v x = δ v,x , and δ v,x is given by ( 3), (15) the equivalent on the left side of ( 12) is simplifying ( 16) we get 1 since ζ∂ q h * g = h * ζ∂ q g, Equation ( 17) can be expressed as Remark 3. Ganesan et al. in [24] and Silverman et al. in [25] demonstrated the following result for q → 1 − and various values of x, y, λ, α and β.
Remark 4. Take note that Theorem 3 makes it simple for us to derive the corresponding condition for a function h ∈ S x,y q (α, β, λ) if and only if where g(ζ) has the form Proof.Let h(ζ) = ζ + wζ v , with satisfying inequality (20).Let g given in Remark 4 and Applying Remark 4, we get h(ζ) ∈ S x,y q (α, β, λ).Conversely, let h(ζ) ∈ S x,y q (α, β, λ) and let Then, using Theorem 3, we obtain The sufficient condition theorem can be driven by using Remark 4.
Proof.We suppose that a function f defined by x,y q,ρ 1 (α, β, λ; h).We only need to prove that f ∈ S x,y q (α, β, λ) to prove Theorem 6.This claim will be proved in the next three steps.Theorem 3 provides us with this where where γ is given by ( 13) and for 0 , where t v is given by (19).So that |t v | ≤ [v] q (1+|β|) |β−γ| .Secondly we obtain that ( 24) is equivalent to ζ , (by using (26)) This proves that Our observations (25) indicate that f ∈ S x,y q (α, β, λ).This concludes the proof of the theorem.

Conclusions
Motivated by recent applications of the theory of (x, y)-symmetrical functions and the concept of q-calculus in geometric function theory, we have employed these two concepts to define and analyze the classes S x,y q (α, β, λ) and K x,y q (α, β, λ) for λ-spiral-like functions in the open unit disk.Specifically, we have explored a convolution property, which will serve as a foundation for further investigation.We have also derived the sufficient condition and equivalent condition.Additionally, we have examined various neighborhood results for the class S x,y q (α, β, λ).The methodology presented in this paper can be readily applied to establish multiple classes with distinct image domains.The utilization of symmetric q-calculus, the Janowski class, and the essential q-hypergeometric functions offers ample opportunities for research in various fields.