Coefﬁcient Estimates for a Subclass of Bi-Univalent Functions Deﬁned by q -Derivative Operator

: Recently, a number of features and properties of interest for a range of bi-univalent and univalent analytic functions have been explored through systematic study, e.g., coefﬁcient inequalities and coefﬁcient bounds. This study examines S δ q ( ϑ , η , ρ , ν ; ψ ) as a novel general subclass of Σ which comprises normalized analytic functions, as well as bi-univalent functions within ∆ as an open unit disk. The study locates estimates for the | a 2 | and | a 3 | Taylor–Maclaurin coefﬁcients in functions of the class which is considered. Additionally, links with a number of previously established ﬁndings are presented.


Introduction
Geometric function theory research has provided analysis of a number of subclasses of A, as a class of normalised analytic function, using a range of approaches. Q-calculus has been widely applied in investigating a number of such subclasses within the open unit disk ∆. ∂ q as a q-derivative operator was initially applied by Ismail et al. [1] in studying a specific q-analogue within ∆ in the starlike function class. Such q-operators were also approximated and their geometric properties examined by Mohammed and Darus [2] for several analytic function subclasses within compact disks. The definition of the q-operators involved was done through convolution normalised analytic and q-hypergeometric functions, and revealed a number of notable findings reported in [3,4]. Raghavendar and Swaminathan [5] studied basic q-close-to-convex function properties, while Aral et al. [6] identified q-calculus applications within operator theory. In addition, fractional q-derivative and fractional q-integral operators, among other q-calculus operators, have been applied in constructing a number of analytic function subclasses, as in [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21].
The function class is denoted by A which represented by the following form: that are analytic in the region ∆ = {ξ ∈ C : |ξ| < 1} and satisfy the following normalization conditions: k(0) = 0 and k (0) = 1.
In addition, let S be the subclass of A consisting of univalent function in ∆.
Especially, if the function h is univalent within ∆, then the above mentioned subordination is comparable to: It should be noted here that Koebe one-quarter theorem as described by [22] stipulates that ∆ images in each univalent function k ∈ A have a disc with a 1/4 radius, meaning that each univalent function k produces k −1 as its inverse, which is characterized as and A function k ∈ A is said to be bi-univalent in ∆ if both k and k −1 are univalent in ∆. Here, Σ represents the bi-univalent function class which Equation (1) defines. Some of the examples of functions within the class Σ are listed here as below (see Srivastava et al. [23]): However, the well known Koebe function is not within the class Σ.
Other common examples of functions within the class S such as ξ − ξ 2 2 and ξ 1 − ξ 2 are also not within the class Σ.
This study begins with definitions of the principal terms used and in-depth concepts for the applications of q-calculus used. In this report, it is assumed that 0 < q < 1. Definitions are first given for fractional q-calculus operators in a complex-valued function k(ξ), as follows: ∑ m−1 n=0 q n = 1 + q + q 2 + · · · + q m−1 , j = m ∈ N.
Definition 3. (see [34,35]) Let k ∈ A and 0 < q < 1. The q-derivative operator of a function k is defined by We note from Definition 3 that From Equations (1) and (4), we get [j] q a j ξ j−1 .
Moreover, as q −→ 1 we have where R δ k(ξ) is Ruscheweyh differential operator which was introduced in [36] and a number of authors have studied it before, see for instance [37,38]. The aim of the present work is to introduce S δ q (ϑ, η, ρ, ν; ψ) as a general subclass of Σ as a class of bi-univalent functions. Within this, estimates are derived for initial coefficients |a 2 | and |a 3 | for functions within the general subclass. Below, various bi-univalent general subclasses are introduced.
, if it is satisfying the following subordination conditions : and where the function χ is given by Equation (2).

Remark 1.
It can clearly be seen that when parameters ϑ, η, ρ, δ, q and ν are specialised, this produces a number of established Σ subclasses, and there are a number of recent works which examine these. Examples are provided for these subclasses.
In order to prove our main results, we need the subsequent lemma.
where P is the family of all analytic functions p in ∆, for which Additionally, some useful work associated with inequalities and their properties can be read in [44][45][46][47].

A Set of Main Results
This section starts by establishing estimates for S δ q (ϑ, η, ρ, ν; ψ) class function for coefficients |a 2 | and |a 3 |.

It follows from Equations
and Clearly, substituting Equations (17) and (18) into Equations (13) and (14), respectively, in the event that we make use of Equation (7), we get and Moreover, and Now, equating the coefficients in Equations (19)-(22), we get and and From Equations (23) and (25), we find that By adding Equations (24) and (26), and then using Equation (27), we obtain For the purpose of brevity, we will utilize the notations given in Equations (10)- (12). Now, making use of the notations defined above and combining Equations (23) and (28), we get Applying Lemma 1 to the coefficients p 2 and h 2 , we find that so that where Ψ(η, ρ, ν, δ, q) and Θ(η, ρ, ν, δ, q) are given by Equations (10) and (12), respectively.

Applications of the Main Result
This part of the paper presents certain distinctive cases within the broader results, provided as corollaries. First of all, by letting in Definition 4 of the class S δ q (ϑ, η, ρ, ν; ψ), we obtain a new class S 1,δ q (ϑ, η, ρ, ν; C, D) given by Definition 5.

Remark 2.
Taking δ = 0 and q → 1 in Theorem 1, we obtain the following result.