Applications of Bernoulli Polynomials and q²-Srivastava–Attiya Operator in the Study of Bi-Univalent Function Classes

Abstract

The central focus of this study is the development and investigation of a generalized subclass of bi-univalent functions, defined using the $q^2$-Srivastava–Attiya operator in conjunction with Bernoulli polynomials. We derive initial coefficient estimates for functions in the newly proposed class and also provide bounds for the Fekete–Szegö functional. In addition to presenting several new findings, we also explore meaningful connections with previously established results in the theory of bi-univalent and subordinate functions, thereby extending and unifying the existing literature in a novel direction.

1. Preliminaries

Geometric function theory (GFT) is a vibrant and evolving field within complex analysis that continues to attract considerable research interest. A central topic in GFT is the study of holomorphic univalent functions and their numerous subclasses. In recent years, growing attention has been directed toward bi-univalent functions due to their intricate geometric properties and the challenging problems they pose in coefficient theory. These function classes offer rich opportunities for exploring functional inequalities, coefficient bounds, and operator-based extensions. This work adds to the expanding research in the area by introducing and analyzing a few subclasses of bi-univalent functions defined via Bernoulli polynomials and $q^2$-calculus techniques.

Define $\mathfrak{U}=$$\{\varsigma \in \mathbb{C}:|\varsigma |<1\}$. Let $\mathcal{A}$ be the class of holomorphic functions $\varphi$ in $\mathfrak{U}$ of the form

$$\varphi \left(\varsigma \right)=\varsigma +d_2{\varsigma }^2+d_3{\varsigma }^3+\cdots =\varsigma +\sum _{j=2}^{\infty }{d}_j{\varsigma }^j,\varsigma \in \mathfrak{U}.$$

(1)

Let $\mathsf{S}$ be the set of all functions $\varphi \in \mathcal{A}$ such that $\varphi$ is univalent in $\mathfrak{U}$. In [1], Bieberbach conjectured that $|d_j|\le j,j\ge 2$ for every function $\varphi \in \mathsf{S}$. Reserchers spent many years trying to prove this conjecture. The Bieberbach conjecture inspired the creation of several subclasses of $\mathsf{S}$, through which many foundational results in geometric function theory were achieved. Finally, Branges solved this Biebereach conjecture for every $j\ge 2$ in [2]. Another problem in GFT is the Fekete–Szegö Functional (FSF) $|d_3-\xi d_2^2|,\xi \in \mathbb{R}$, for every function $\varphi \in \mathsf{S}$ [3]. The aforementioned problem has been extensively studied by prominent researchers for functions in various subclasses of $\mathsf{S}$. Among the various subclasses of $\mathsf{S}$, the class $\sigma$ of bi-univalent functions has attracted significant attention. The concept of bi-univalent functions, forming the class $\sigma$, was introduced by Lewin in [4]. In this context, $\varphi$ is an analytic function such that both $\varphi$ and its inverse ${\varphi }^{-1}=\psi$ are univalent in $\mathfrak{U}$. Every function $\varphi \in \mathsf{S}$ of the form (1) has an inverse given by the well-known Koebe theorem (see [5]),

$${\varphi }^{-1}\left(w\right)=w-d_2{w}^2+(2d_2^2-d_3)w^3-(5d_2^3-5d_2{d}_3+d_4)w^4+\cdots =\psi \left(w\right)$$

(2)

such that $\varsigma =\psi \left(\varphi \right(\varsigma \left)\right)$ and $w=\varphi \left(\psi \left(w\right)\right),\left|w\right|<r_0\left(\varphi \right),1/4\le r_0\left(\varphi \right),\varsigma ,w\in \mathfrak{U}$. The class $\sigma$ is non-empty, as evidenced by functions such as ${\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\varsigma }{1-\varsigma }}\right)$, ${\displaystyle \frac{\varsigma }{1-\varsigma }}$, and $-log(1-\varsigma )$, all of which belong to the family $\sigma$. However, ${\displaystyle \frac{2\varsigma -{\varsigma }^2}{2}}$, ${\displaystyle \frac{\varsigma }{1-{\varsigma }^2}}$, and the Koebe function ${\displaystyle \frac{\varsigma }{{(1-\varsigma )}^2}}$ are not elements of $\sigma$, even though they are in $\mathsf{S}$. For a brief but informative look at the class $\sigma$ and some of its properties, refer to [6,7,8,9]. Research on the family of bi-univalent functions has recently gained momentum thanks to an article by Srivastava and his co-authors [10]. Since this article revived the topic, numerous researchers have looked into a number of intriguing special families of $\sigma$; see [11,12,13,14] and the citations given in these papers.

Here, we review the q-derivative operator.

Definition 1

([15]). Let $\mathsf{\Phi }: \mathbb{C}\to \mathbb{C}$ be a function. Then, the q-derivative of $\mathsf{\Phi }$, denoted by $D_q\mathsf{\Phi }\left(\varsigma \right)$, is defined by

$$D_q\mathsf{\Phi }\left(\varsigma \right)={\displaystyle \frac{\mathsf{\Phi }\left(\varsigma \right)-\mathsf{\Phi }\left(q\varsigma \right)}{(1-q)\varsigma }}(\varsigma \in \mathbb{C}\setminus \left\{0\right\}),0<q<1,$$

and $D_q\mathsf{\Phi }\left(0\right)={\mathsf{\Phi }}^{\prime }\left(0\right)$, provided ${\mathsf{\Phi }}^{\prime }\left(0\right)$ exists.

We note that the q-bracket number, denoted by ${\left[j\right]}_q$, is defined by ${\left[j\right]}_q=1+q+q^2+\dots +q^{j-1}={\displaystyle \frac{1-q^j}{1-q}},(q\ne 1)$. One can easily verify that $D_q{\varsigma }^j={\left[j\right]}_q{\varsigma }^{j-1}$ and $D_{q}ln\left(\varsigma \right)={\displaystyle \frac{ln(1/q)}{(1-q)\varsigma }}$. Also, we observe that ${\left[j\right]}_q\to j,ifq\to 1^{-}$. Therefore, $D_q\mathsf{\Phi }\left(\varsigma \right)\to {\mathsf{\Phi }}^{\prime }\left(\varsigma \right)$ as $q\to 1^{-}$.

Definition 2

([16]). Let $\lambda \in \mathbb{C}$ and $0<q<1$. The symmetric q-number denoted by $\widetilde{{\left[\lambda \right]}_q}$ is defined as $\widetilde{{\left[\lambda \right]}_q}={\displaystyle \frac{q^{\lambda }-q^{-\lambda }}{q-q^{-1}}}$.

For $\lambda =j\in \mathbb{N}\cup \left\{0\right\}$, the symmetric q-number $\widetilde{{\left[j\right]}_q}$ is defined as

$$\widetilde{{\left[j\right]}_q}={\displaystyle \frac{q^j-q^{-j}}{q-q^{-1}}}=\left\{\begin{array}{cc}{\displaystyle \frac{1}{q^{j-1}}}(1+q^2+\cdots +q^{2j-2})\hfill & ;ifj\ne 0\hfill \\ 0\hfill & ;ifj=0.\hfill \end{array}\right..$$

It is straightforward to observe that $li{m}_{q\to 1^{-}}{\left[j\right]}_{q^2}$=j. It is also important to note that the expression ${\left[j\right]}_{q^2}$, which frequently appears in the study of the q-deformed quantum mechanical simple harmonic oscilator [17], does not reduce to ${\left[j\right]}_q$. For $j\in \mathbb{N}\cup \left\{0\right\}$ and $0<q<1$, the $q^2$-number satisfies the following identities:

$${[j+1]}_{q^2}=q^j+{\displaystyle \frac{{\left[j\right]}_{q^2}}{q}},and{\left[j\right]}_{q^2}={\displaystyle \frac{{[j+b]}_{q^2}{q}^j-{\left[b\right]}_{q^2}}{q^{j+b}}}(b\in \mathbb{N}).$$

(3)

In [18], it was noted that the so-called “symmetric” q-number $\widetilde{{\left[\lambda \right]}_q}$, as defined in Definition 2, is merely a trivial and inconsequential variation of the q-number ${\left[\lambda \right]}_{q^2}$ by multiplying the later by $q^{1-\lambda }$. Consequently, they adopted the notation ${\left[\lambda \right]}_{q^2}$ in place of $\widetilde{{\left[\lambda \right]}_q}$, referring to it as the $q^2$-number.

We now define the $q^2$-derivative.

Definition 3

([16]). Let $\mathsf{\Phi }$: $\mathbb{C}\to \mathbb{C}$ be a function. Then, the $q^2$-derivative of $\mathsf{\Phi }$, denoted by $D_{q^2}\mathsf{\Phi }\left(\varsigma \right)$, is defined as

$$D_{q^2}\mathsf{\Phi }\left(\varsigma \right)={\displaystyle \frac{\mathsf{\Phi }\left(q\varsigma \right)-\mathsf{\Phi }\left(q^{-1}\varsigma \right)}{(q-q^{-1})\varsigma }}(\varsigma \in \mathbb{C}\setminus \left\{0\right\}),$$

and $D_{q^2}\mathsf{\Phi }\left(0\right)={\mathsf{\Phi }}^{\prime }\left(0\right)$, provided ${\mathsf{\Phi }}^{\prime }\left(0\right)$ exists.

We note that $D_{q^2}{\varsigma }^j$=${\left[j\right]}_{q^2}{\varsigma }^{j-1}$, and hence, for $\varphi \left(\varsigma \right)$ of the form (1),

$$D_{q^2}\varphi \left(\varsigma \right)=1+\sum _{j=2}^{\infty }{\left[j\right]}_{q^2}{d}_j{\varsigma }^{j-1}.$$

(4)

For ${\mathfrak{a}}_{\mathfrak{1}}$, ${\mathfrak{a}}_{\mathfrak{2}}$$\in \mathcal{A}$, both holomorphic in $\mathfrak{U}$, the function ${\mathfrak{a}}_{\mathfrak{1}}$ is said to be subordinate to ${\mathfrak{a}}_{\mathfrak{2}}$, denoted by ${\mathfrak{a}}_{\mathfrak{1}}\prec {\mathfrak{a}}_{\mathfrak{2}}$ or ${\mathfrak{a}}_{\mathfrak{1}}\left(\varsigma \right)\prec {\mathfrak{a}}_{\mathfrak{2}}\left(\varsigma \right)$, if there is a Schwarz function $\theta \left(\varsigma \right)$ that is holomorphic in $\varsigma \in \mathfrak{U}$ and satisfies $\theta \left(0\right)=0$ and $\left|\theta \right(\varsigma \left)\right|<1$, for all $\varsigma \in \mathfrak{U}$, such that

$${\mathfrak{a}}_{\mathfrak{1}}\left(\varsigma \right)={\mathfrak{a}}_{\mathfrak{2}}\left(\theta \left(\varsigma \right)\right),\varsigma \in \mathfrak{U}.$$

Furthermore, if ${\mathfrak{a}}_{\mathfrak{2}}\in \mathsf{S}$, then

$${\mathfrak{a}}_{\mathfrak{1}}\left(\varsigma \right)\prec {\mathfrak{a}}_{\mathfrak{2}}\left(\varsigma \right)\iff {\mathfrak{a}}_{\mathfrak{1}}\left(0\right)={\mathfrak{a}}_{\mathfrak{2}}\left(0\right)and{\mathfrak{a}}_{\mathfrak{1}}\left(\mathfrak{U}\right)\subset {\mathfrak{a}}_{\mathfrak{2}}\left(\mathfrak{U}\right).$$

Special polynomials such as Chebyshev, Fibonacci, Horadam, Laguerre, Hermite, and Legendre, and their numerous generalizations, have been extensively studied due to their deep theoretical significance and wide-ranging applications in areas including number theory, combinatorics, physics, and engineering. Motivated by the broad utility of such polynomials, the present work focuses on a specific subfamily of the analytic function class $\sigma$, which is intimately linked to classical number sequences and special polynomials. This subfamily has been the subject of growing interest, particularly concerning coefficient bounds and the Fekete–Szegö functional (FSF), as studied in the recent literature (see [19,20,21]). Notably, Bernoulli polynomials have emerged as a key focus in this context, yielding significant results for subclasses of $\sigma$ and highlighting new avenues for exploration within geometric function theory [22,23].

The Bernoulli polynomials $B_j\left(x\right),x\in \mathbb{R}$, and j are non-negative integers, and are frequently specified (see, [24]) using the generating function:

$$\mathfrak{B}(x,\varsigma )={\displaystyle \frac{\varsigma e^{x\varsigma }}{e^{\varsigma }-1}}=B_0\left(x\right)+B_1\left(x\right)\varsigma +B_2\left(x\right){\displaystyle \frac{{\varsigma }^2}{2!}}+\cdots .\left|\varsigma \right|<2|\pi .$$

(5)

With the following recursion, the Bernoulli polynomials can be easily calculated:

$$\sum _{n=0}^{j-1}\left(\begin{array}{c}j\\ n\end{array}\right)B_n\left(x\right)=j{x}^{j-1},j=2,3,4,\cdots ,$$

with the initial condition $B_0\left(x\right)=1$. The following are the first few Bernoulli polynomials:

$$B_1\left(x\right)={\displaystyle \frac{2x-1}{2}},B_2\left(x\right)=x(x-1)+{\displaystyle \frac{1}{6}},\dots .$$

(6)

The recent survey and expository review article [25] offers an in-depth and systematic examination of the Hurwitz–Lerch zeta function $f(s,b;\varsigma )$, elucidating its key properties, analytic structure, and functional characteristics. The general form of the Hurwitz–Lerch zeta function is given by $f(s,b;\varsigma )$=${\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{\varsigma }^j}{{(j+b)}^s}}$ ($s\in \mathbb{C}$ when $\left|\varsigma \right|<1,\Re e\left(s\right)>0$ when $\left|\varsigma \right|=1$, and $b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-})$. This function admits a meromorphic continuation to the entire complex s-plane, with a simple pole at s = 1 having residue 1. Consider the q-analogue of the Hurwitz–Lerch zeta function [26] given by the following series: $f_q(s,b;\varsigma )$ = ${\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{\varsigma }^j}{{[j+b]}_q^s}}$, where $s\in \mathbb{C}$ when $\left|\varsigma \right|<1,\Re e\left(s\right)>0$ when $\left|\varsigma \right|=1$, and $b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$. For $\varphi \in \mathcal{A}$, the q-analogue of the Srivastava–Attiya operator [26] is given by

$$J_{q,b}^s\varphi \left(\varsigma \right)=g_q(s,b;\varsigma )\ast \varphi \left(\varsigma \right)=\varsigma +\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{[1+b]}_q}{{[j+b]}_q}}\right)}^s{d}_j{\varsigma }^j,$$

where

$$g_q(s,b;\varsigma )={[1+b]}_q^s\left(f_q(s,b;\varsigma )-{\displaystyle \frac{1}{{\left[b\right]}_q^s}}\right)=\varsigma +\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{[1+b]}_q}{{[j+b]}_q}}\right)}^s{\varsigma }^j;$$

$s\in \mathbb{C}$ when $\left|\varsigma \right|<1,\Re e\left(s\right)>0$ when $\left|\varsigma \right|=1$, and $b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$. The q-Srivastava-Attiya operator $J_{q,b}^s$, which has been widely and extensively studied, serves as a unifying generalization of several well-known operators previously investigated in the literature and finds numerous applications in geometric function theory [27].

In [18], the authors introduced the $q^2$-analogue of the well-known Hurwitz–Lerch zeta function, defined as

$$f_{q^2}(s,b;\varsigma )=\sum _{j=0}^{\infty }{\displaystyle \frac{{\varsigma }^j}{{[j+b]}_{q^2}^s}},$$

which upon normalizing gives

$$g_{q^2}(s,b;\varsigma )={[1+b]}_{q^2}^s\left(f_{q^2}(s,b;\varsigma )-{\displaystyle \frac{1}{{\left[b\right]}_{q^2}^s}}\right)=\varsigma +\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{[1+b]}_{q^2}}{{[j+b]}_{q^2}}}\right)}^s{\varsigma }^j,$$

(7)

where $s\in \mathbb{C}$ when $\left|\varsigma \right|<1,\Re e\left(s\right)>0$ when $\left|\varsigma \right|=1$, and $b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$. By utilizing the series representations given in Equations (1) and (7), the authors defined the $q^2$ -Srivastava–Attiya operator $J_{q^2,b}^s:\mathcal{A}\to \mathcal{A}$ as follows:

$$J_{q^2,b}^s\varphi \left(\varsigma \right)=g_{q^2}(s,b;\varsigma )\ast \varphi \left(\varsigma \right)=\varsigma +\sum _{j=2}^{\infty }{\left({\displaystyle \frac{{[1+b]}_{q^2}}{{[j+b]}_{q^2}}}\right)}^s{d}_j{\varsigma }^j.$$

(8)

Remark 1.

Taking the limit as $q\to 1^{-}$, the $q^2$-Srivastava–Attiya operator $J_{q^2,b}^s\varphi \left(\varsigma \right)$ reduces to the classical Srivastava–Attiya operator, as discussed in [28]. By restricting b to the set of natural numbers and employing Equations (4) and (8), one readily obtains the following identity:

$$\varsigma D_{q^2}{J}_{q^2,b}^{s+1}\varphi \left(\varsigma \right)=\left(1+{\displaystyle \frac{{\left[b\right]}_{q^2}}{q^{b+1}}}\right)J_{q^2,b}^s\varphi \left(\varsigma \right)-q\left({\displaystyle \frac{{\left[b\right]}_{q^2}}{q^{b+1}}}{J}_{q^2,b}^{s+1}\varphi \left(q^{-1}\varsigma \right)\right).$$

(9)

The identity (9) reduces to $\varsigma J_{s+1,b}^{\prime }\varphi \left(\varsigma \right)=(1+b)J_{s,b}\varphi \left(\varsigma \right)-b{J}_{s+1,b}\varphi \left(\varsigma \right)$, which was originally established in [29].

Remark 2.

The $q^2$-Srivastava–Attiya operator serves as a natural and effective tool in this context due to its unifying and generalizing capacity within the theory of analytic and bi-univalent functions. This operator incorporates elements of q-calculus, a framework that generalizes classical analysis and has shown rich connections with special functions, orthogonal polynomials, and operator theory. By involving the $q^2$-parameter, the operator introduces an additional layer of flexibility and control over the behavior of analytic functions, allowing for finer tuning in the study of coefficient estimates and functional inequalities.

More specifically, the Srivastava–Attiya operator has been previously shown to generate broad subclasses of analytic functions with significant structural properties. Its integration with q-calculus enables the exploration of function classes that interpolate between known results in classical analysis and more generalized or quantum-calculus-based frameworks. This makes it particularly well-suited for studying functions associated with special polynomials like the Bernoulli and Euler families, which themselves have q-analogues and rich algebraic structures.

In the study of bi-univalent functions particularly for bounding Maclaurin coefficients and the Fekete–Szegö functional, the $q^2$-Srivastava–Attiya operator provides a versatile and well-integrated framework compatible with classical geometric function theory techniques. Its ability to unify classical results with modern generalizations naturally motivates the definition of the subclass ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$ and its connection to special polynomials. Building on this, we introduce a comprehensive subfamily of $\sigma$ associated with the $q^2$-Srivastava–Attiya operator governed by Bernoulli polynomials.

This paper employs the function $\mathfrak{B}(x,\varsigma )$ as stated in (5), $\psi \left(w\right)={\varphi }^{-1}\left(w\right)$ as stated in (2), $\alpha \in \mathbb{C}\setminus \left\{0\right\}$, $0<q<1$, $b\in \mathbb{C}\setminus {\mathbb{Z}}_0^{-}$, $s\in \mathbb{C}$, and $\varsigma ,w\in \mathfrak{U}$, unless otherwise noted.

Definition 4.

Let $\tau \ge 1$, $0\le \beta \le 1$, and $0\le \gamma \le 1$. If $\varphi \in \sigma$ satisfies

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)+(1-\gamma )\left({\displaystyle \frac{\varsigma {\left(D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)}^{\tau }}{(1-\beta )\varsigma +\beta J_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)-1\right)\prec \mathfrak{B}(x,\varsigma ),$$

(10)

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)}}\right)+(1-\gamma )\left({\displaystyle \frac{w{\left(D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)}^{\tau }}{(1-\beta )w+\beta J_{q^2,b}^s\psi \left(w\right)}}\right)-1\right)\prec$$

$$\mathfrak{B}(x,w),$$

(11)

which means we can say that $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$.

By assigning specific values to $\tau$, $\beta$, and $\gamma$, one can derive distinct and structurally significant subfamilies, from the family ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$, each possessing unique geometric properties.

$\mathbf{Case}\mathbf{1}$. If $\tau =1$, then ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,1)$ is a subfamily of elements $\varphi \in \sigma$ satisfying

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{D_{q^2}\left(\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)}{1-\beta +\beta D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)+(1-\gamma )\left({\displaystyle \frac{\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)}{(1-\beta )\varsigma +\beta J_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)-1\right)\prec \mathfrak{B}(x,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{D_{q^2}\left(w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)}{1-\beta +\beta D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)}}\right)+(1-\gamma )\left({\displaystyle \frac{w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)}{(1-\beta )w+\beta J_{q^2,b}^s\psi \left(w\right)}}\right)-1\right)\prec$$

$$\mathfrak{B}(x,w),$$

where $0\le \gamma \le 1$, and $0\le \beta \le 1$.

$\mathbf{Case}\mathbf{2}$. If $\gamma =0$, then we have a subfamily ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,\beta ,\tau )$ of functions such that $\varphi \in \sigma$, satisfying

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{\varsigma {\left(D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)}^{\tau }}{(1-\beta )\varsigma +\beta J_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)-1\right)\prec \mathfrak{B}(x,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{w{\left(D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)}^{\tau }}{(1-\beta )w+\beta J_{q^2,b}^s\psi \left(w\right)}}\right)-1\right)\prec \mathfrak{B}(x,w),$$

where $\tau \ge 1$, and $0\le \beta \le 1$.

$\mathbf{Case}\mathbf{3}$. If $\gamma =1$, then we have a subfamily ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,\beta ,\tau )$ of elements $\varphi \in \sigma$, satisfying

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{{\left[D_{q^2}\left(\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)-1\right)\prec \mathfrak{B}(x,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{{\left[D_{q^2}\left(w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)}}\right)-1\right)\prec \mathfrak{B}(x,w),$$

where $0\le \beta \le 1$, and $\tau \ge 1$.

$\mathbf{Case}\mathbf{4}$. Setting $\beta =0$ yields a new subclass, denoted by ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,0,\tau )$, consisting of elements $\varphi \in \sigma$ that satisfy the following conditions:

$$1+{\displaystyle \frac{1}{\alpha }}(\gamma {\left[D_{q^2}\left(\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)\right]}^{\tau }+(1-\gamma ){\left(D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)}^{\tau }-1)\prec \mathfrak{B}(x,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}(\gamma {\left[D_{q^2}\left(w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)\right]}^{\tau }+(1-\gamma ){\left(D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)}^{\tau }-1)\prec \mathfrak{B}(x,w),$$

where $0\le \gamma \le 1$, and $\tau \ge 1$.

The following particular important instances from the family ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,0,\tau )$ are obtained by taking $\gamma =0$ and $\gamma =1$.

$\mathbf{Instance}\mathbf{1}$. Let $\tau \ge 1$. If a function $\varphi \in \sigma$ satisfies

$${\displaystyle \frac{1}{\alpha }}({\left(D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)}^{\tau }-1)+1\prec \mathfrak{B}(x,\varsigma ),and{\displaystyle \frac{1}{\alpha }}({\left(D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)}^{\tau }-1)+1\prec \mathfrak{B}(x,w),$$

then we say that $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,0,\tau )$.

$\mathbf{Instance}\mathbf{2}$. Let $\tau \ge 1$. If a function $\varphi \in \sigma$ satisfies

$${\displaystyle \frac{1}{\alpha }}({\left[D_{q^2}\left(\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)\right]}^{\tau }-1)+1\prec \mathfrak{B}(x,\varsigma ),and{\displaystyle \frac{1}{\alpha }}({\left[D_{q^2}\left(w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)\right]}^{\tau }-1)+1\prec \mathfrak{B}(x,w),$$

then we say that $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,0,\tau )$.

$\mathbf{Case}\mathbf{5}$. Letting $\beta =1$ yields a new subclass, denoted by ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,1,\tau )$, consisting of functions $\varphi \in \sigma$ that satisfy the following conditions:

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)\right]}^{\tau }}{D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)+(1-\gamma )\left({\displaystyle \frac{\varsigma {\left(D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)}^{\tau }}{J_{q^2,b}^s\varphi \left(\varsigma \right)}}\right)-1\right)\prec \mathfrak{B}(x,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)\right]}^{\tau }}{D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)}}\right)+(1-\gamma )\left({\displaystyle \frac{w{\left(D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)}^{\tau }}{J_{q^2,b}^s\psi \left(w\right)}}\right)-1\right)\prec \mathfrak{B}(x,w),$$

where $0\le \gamma \le 1$, and $\tau \ge 1$.

Taking $\gamma =0$ and $\gamma =1$ yields the following specific important instances from the family ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,1,\tau )$.

$\mathbf{Instance}\mathbf{3}$. Let $\tau \ge 1$. If a function $\varphi \in \sigma$ satisfies

$$1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{\varsigma {\left(D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)}^{\tau }}{J_{q^2,b}^s\varphi \left(\varsigma \right)}}-1\right)\prec \mathfrak{B}(x,\varsigma ),and1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{w{\left(D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)}^{\tau }}{J_{q^2,b}^s\psi \left(w\right)}}-1\right)\prec \mathfrak{B}(x,w),$$

then we say that $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,1,\tau )$.

$\mathbf{Instance}\mathbf{4}$. Let $\tau \ge 1$. If a function $\varphi \in \sigma$ satisfies

$$1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{{\left[D_{q^2}\left(\varsigma D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)\right)\right]}^{\tau }}{D_{q^2}{J}_{q^2,b}^s\varphi \left(\varsigma \right)}}-1\right)\prec \mathfrak{B}(x,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{{\left[D_{q^2}\left(w{D}_{q^2}{J}_{q^2,b}^s\psi \left(w\right)\right)\right]}^{\tau }}{D_{q^2}{J}_{q^2,b}^s\psi \left(w\right)}}-1\right)\prec \mathfrak{B}(x,w),$$

then we say that $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,1,\tau )$.

The following is the structure of the paper’s content. For functions in the family ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$, the estimates for $|d_2|$, $|d_3|$, and $|d_3-\xi d_2^2|,\xi \in \mathbb{R}$, are found in Section 2. In Section 3, we highlight relevant connections between some of the particular cases and the key conclusions. We also go over some observations regarding our findings. In Section 4, we conclude the study with some observations.

2. Key Findings

Section 2 commences with the derivation of bounds for the initial Maclaurin’s coefficients and $|d_3-\xi d_2^2|,\xi \in \mathbb{R}$, for any function $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$.

Theorem 1.

Let $\tau \ge 1$, $0\le \beta \le 1$ and $0\le \gamma \le 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3+T{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

(12)

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}},$$

(13)

and for $\xi \in \mathbb{R}$

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}\hfill & ;|1-\xi |\le \daleth \hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3+T{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge \daleth ,\hfill \end{array}\right.\end{array}$$

(14)

where

$$\daleth =\left|{\displaystyle \frac{((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3+T{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha (\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3{(2x-1)}^2}}\right|,$$

(15)

$$T={\displaystyle \frac{\tau (\tau -1){\left[2\right]}_{q^2}^2}{2}}-\tau \beta {\left[2\right]}_{q^2}+{\beta }^2,A_2={\left({\displaystyle \frac{{[1+b]}_{q^2}}{{[2+b]}_{q^2}}}\right)}^{s}and{A}_3={\left({\displaystyle \frac{{[1+b]}_{q^2}}{{[3+b]}_{q^2}}}\right)}^s,$$

(16)

$${\gamma }_1=1-\gamma +\gamma {\left[2\right]}_{q^2},{\gamma }_2=1-\gamma +\gamma {\left[2\right]}_{q^2}^2,and{\gamma }_3=1-\gamma +\gamma {\left[3\right]}_{q^2}.$$

(17)

Proof. 

Let $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$. Then, from subordinations (10) and (11), we can write

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(\varsigma D_{q^2}\varphi \left(\varsigma \right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}\varphi \left(\varsigma \right)}}\right)+(1-\gamma )\left({\displaystyle \frac{\varsigma {\left(D_{q^2}\varphi \left(\varsigma \right)\right)}^{\tau }}{(1-\beta )\varsigma +\beta \varphi \left(\varsigma \right)}}\right)-1\right)=\mathfrak{B}(x,\mathfrak{l}\left(\varsigma \right)),$$

(18)

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(w{D}_{q^2}\psi \left(w\right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}\psi \left(w\right)}}\right)+(1-\gamma )\left({\displaystyle \frac{w{D}_{q^2}{\left(\psi \left(w\right)\right)}^{\tau }}{(1-\beta )w+\beta \psi \left(w\right)}}\right)-1\right)=\mathfrak{B}(x,\mathfrak{m}\left(w\right)),$$

(19)

where Schwarz functions $\mathfrak{l}\left(\varsigma \right)={\mathfrak{l}}_1\varsigma +{\mathfrak{l}}_2{\varsigma }^2+\cdots ,\varsigma \in \mathfrak{U}$, and $\mathfrak{m}\left(w\right)={\mathfrak{m}}_{1}w+{\mathfrak{m}}_2{w}^2+\cdots ,w\in \mathfrak{U}$, satisfy (see [5])

$$|{\mathfrak{l}}_j|\le 1,and|{\mathfrak{m}}_j|\le 1(j\in \mathbb{N}).$$

(20)

Using standard mathematical techniques, Equation (18) can be rewritten as follows:

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(\varsigma D_{q^2}\varphi \left(\varsigma \right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}\varphi \left(\varsigma \right)}}\right)+(1-\gamma )\left({\displaystyle \frac{\varsigma D_{q^2}{\left(\varphi \left(\varsigma \right)\right)}^{\tau }}{(1-\beta )\varsigma +\beta \varphi \left(\varsigma \right)}}\right)-1\right)$$

$$=1+{\displaystyle \frac{1}{\alpha }}(\tau {\left[2\right]}_{q^2}-\beta ){\gamma }_1{A}_2{d}_2\varsigma +{\displaystyle \frac{1}{\alpha }}\left((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3{d}_3+T{\gamma }_2{A}_2^2{d}_2^2\right){\varsigma }^2+\cdots ,$$

(21)

and

$$\mathfrak{B}(x,\mathfrak{l}\left(\varsigma \right))=1+B_1\left(x\right){\mathfrak{l}}_1\varsigma +\left(B_1\left(x\right){\mathfrak{l}}_2+{\displaystyle \frac{B_2\left(x\right)}{2!}}{\mathfrak{l}}_1^2\right){\varsigma }^2+\cdots ,$$

(22)

where T, $A_2$ and $A_3$ are as mentioned in (16), and ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ are as stated in (17). Using standard mathematical techniques, Equation (19) can be rewritten as follows:

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[D_{q^2}\left(w{D}_{q^2}\psi \left(w\right)\right)\right]}^{\tau }}{1-\beta +\beta D_{q^2}\psi \left(w\right)}}\right)+(1-\gamma )\left({\displaystyle \frac{w{\left(D_{q^2}\psi \left(w\right)\right)}^{\tau }}{(1-\beta )w+\beta \psi \left(w\right)}}\right)-1\right)$$

$$=1-{\displaystyle \frac{1}{\alpha }}(\tau {\left[2\right]}_{q^2}-\beta ){\gamma }_1{A}_2{d}_{2}w+{\displaystyle \frac{1}{\alpha }}\left((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3(2d_2^2-d_3)+T{\gamma }_2{A}_2^2{d}_2^2\right)w^2+\cdots ,$$

(23)

and

$$\mathfrak{B}(x,\mathfrak{m}\left(w\right))=1+B_1\left(x\right){\mathfrak{m}}_{1}w+\left(B_1\left(x\right){\mathfrak{m}}_2+{\displaystyle \frac{B_2\left(x\right)}{2!}}{\mathfrak{m}}_1^2\right)w^2+\cdots ,$$

(24)

where T, $A_2$ and $A_3$ are as mentioned in (16), and ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ are as stated in (17).

Invoking the equality in (18), we equate the coefficients of like powers in Equations (21) and (22), leading to the following result:

$$(\tau {\left[2\right]}_{q^2}-\beta ){\gamma }_1{A}_2{d}_2=\alpha B_1\left(x\right){\mathfrak{l}}_1,$$

(25)

$$(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3{d}_3+T{\gamma }_2{A}_2^2{d}_2^2=\alpha \left[B_1\left(x\right){\mathfrak{l}}_2+{\displaystyle \frac{B_2\left(x\right)}{2!}}{\mathfrak{l}}_1^2\right],$$

(26)

In a similar manner, equality (19) enables the comparison of terms of the same degree in (23) and (24), from which the following conclusion is drawn:

$$-(\tau {\left[2\right]}_{q^2}-\beta ){\gamma }_1{A}_2{d}_2=\alpha B_1\left(x\right){\mathfrak{m}}_1,$$

(27)

and

$$(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3(2d_2^2-d_3)+T{\gamma }_2{A}_2^2{d}_2^2=\alpha \left[B_1\left(x\right){\mathfrak{m}}_2+{\displaystyle \frac{B_2\left(x\right)}{2!}}{\mathfrak{m}}_1^2\right].$$

(28)

From (25) and (27), we obtain

$${\mathfrak{l}}_1=-{\mathfrak{m}}_1,$$

(29)

and

$$2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2{d}_2^2={\alpha }^2({\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2)B_1^2\left(x\right).$$

(30)

The addition of (26) and (28) yields

$$2((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3+T{\gamma }_2{A}_2^2)d_2^2=\alpha B_1\left(x\right)({\mathfrak{l}}_2+{\mathfrak{m}}_2)+\alpha {\displaystyle \frac{B_2\left(x\right)}{2}}({\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2).$$

(31)

Replacing ${\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2$ from (30) in (31) we get

$$d_2^2={\displaystyle \frac{{\alpha }^2{B}_1^3\left(x\right)({\mathfrak{l}}_2+{\mathfrak{m}}_2)}{2\left[((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3+T{\gamma }_2{A}_2^2)\alpha B_1^2\left(x\right)-{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2\left(\frac{B_2\left(x\right)}{2}\right)\right]}}.$$

(32)

Utilizing (6) for $B_1\left(x\right)$, $B_2\left(x\right)$ and applying (20) to ${\mathfrak{l}}_2$,${\mathfrak{m}}_2$ yields (12).

By subtracting (28) from (26), we obtain the following bound on $|d_3|$:

$$d_3=d_2^2+{\displaystyle \frac{\alpha B_1\left(x\right)({\mathfrak{l}}_2-{\mathfrak{m}}_2)}{2(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}.$$

(33)

If we replace $d_2^2$ from (30) in (33), we obtain

$$|d_3|={\displaystyle \frac{{\alpha }^2{B}_1^2\left(x\right)({\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2)}{2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2}}+{\displaystyle \frac{\alpha B_1\left(x\right)({\mathfrak{l}}_2-{\mathfrak{m}}_2)}{2(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}.$$

(34)

Using (6) and (20), we derive (13) from (34).

Finally, substituting the values of $d_2^2$ and $d_3$ from (32) and (33), respectively, we determine the bound for $|d_3-\xi d_2^2|$. As a result, we obtain

$$|d_3-\xi d_2^2|={\displaystyle \frac{\left|\alpha \right||B_1\left(x\right)|}{2}}\left|\left({\displaystyle \frac{1}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}+\mathfrak{B}\right){\mathfrak{l}}_2-\left({\displaystyle \frac{1}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}-\mathfrak{B}\right){\mathfrak{m}}_2\right|$$

where

$$\mathfrak{B}={\displaystyle \frac{\alpha (1-\xi )B_1^2\left(x\right)}{((\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3+T{\gamma }_2{A}_2^2)\alpha B_1^2\left(x\right)-{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2\left(\frac{B_2\left(x\right)}{2}\right)}}.$$

Clearly

$$|d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right||B_1\left(x\right)|}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}\hfill & ;\left|\mathfrak{B}\right|\le {\displaystyle \frac{1}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}\hfill \\ \left|\alpha \right||B_1\left(x\right)\left|\right|\mathfrak{B}|\hfill & ;\left|\mathfrak{B}\right|\ge {\displaystyle \frac{1}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}.\hfill \end{array}\right.$$

(35)

We derive (14) from (35), with ℸ being the same as in (15). □

Theorem 1 yields the following result for $\xi =1$.

Corollary 1.

If a function $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$, then $|d_3-d_2^2|\le {\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3}}$.

Remark 3.

As $q\to 1^{-}$, the $q^2$-numbers ${\left[n\right]}_{q^2}$ tend toward the classical integers n, and the $q^2$-Srivastava–Attiya operator converges to the classical Srivastava–Attiya operator. Consequently, the class ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$ reduces to a corresponding class defined by the classical Srivastava–Attiya operator governed by Bernoulli polynomials. In this limiting case, the coefficient estimates given in Theorem 1 reduce to their classical counterparts (with all ${\left[n\right]}_{q^2}\to n$, and $A_n\to {\left({\displaystyle \frac{1+b}{n+b}}\right)}^s$, etc.). Thus, the results presented here generalize and unify earlier results in the classical setting.

3. Particular Cases

The generalized bi-univalent function class governed by Bernoulli polynomials encompasses several notable subclasses that emerge through appropriate choices of the parameters $\tau$, $\gamma$, and $\beta$. By assigning specific values to these parameters, one can derive distinct and structurally significant subfamilies, each possessing unique geometric properties. Given the apparent lack of existing literature addressing such classes in this framework, we devote the following section to a detailed investigation of these subclasses.

When $\tau =1$, Theorem 1 yields the following result.

Corollary 2.

Let $0\le \gamma \le 1$, and $0\le \beta \le 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,1)$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|(({\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3-\beta \left({\left[2\right]}_{q^2}\right){\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{({\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha a\right|}^2}{{({\left[2\right]}_q-\beta )}^2{\gamma }_1^2}}+{\displaystyle \frac{\left|\alpha a\right|}{({\left[3\right]}_q-\beta ){\gamma }_3}},$$

and for $\xi \in \mathbb{R}$

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{({\left[3\right]}_q-\beta ){\gamma }_3{A}_3}}\hfill & ;|1-\xi |\le {\daleth }_1\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{\left|a\right|}^3|1-\xi |}{2|(({\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3-\beta \left({\left[2\right]}_{q^2}\right){\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{({\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_1,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_1=\left|{\displaystyle \frac{(({\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3-\beta ({\left[2\right]}_{q^2}-\beta ){\gamma }_2{A}_2^2)\alpha a^2-2{({\left[2\right]}_{q^2}-\beta )}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})}{({\left[3\right]}_{q^2}-\beta ){\gamma }_3{A}_3\alpha {(2x-1)}^2}}\right|,$$

$A_2$ and $A_3$ are as stated in (16), and ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ are as stated in (17).

The outcome of Theorem 1 would be as follows if $\gamma =0$.

Corollary 3.

Let $\tau \ge 1$, and $0\le \beta \le 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,\beta ,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|((\tau {\left[3\right]}_{q^2}-\beta )A_3+T{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{(\tau {\left[2\right]}_{q^2}-\beta )}^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-\beta )A_3}},$$

and for $\xi \in \mathbb{R}$

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-\beta )A_3}}\hfill & ;|1-\xi |\le {\daleth }_2\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|((\tau {\left[3\right]}_{q^2}-\beta )A_3+T{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_2,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_2=\left|{\displaystyle \frac{((\tau {\left[3\right]}_{q^2}-\beta )A_3+T{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha (\tau {\left[3\right]}_{q^2}-\beta )A_3{(2x-1)}^2}}\right|,$$

and T, $A_2$ and $A_3$ are as mentioned in (16).

The following would result if $\gamma =1$, according to Theorem 1:

Corollary 4.

Let $\tau \ge 1$, and $0\le \beta \le 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,\beta ,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|((\tau {\left[3\right]}_{q^2}-\beta ){\left[3\right]}_{q^2}{A}_3+T{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\left[2\right]}_{q^2}^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-\beta ){\left[3\right]}_{q^2}{A}_3}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-\beta ){\left[3\right]}_{q^2}{A}_3}}\hfill & ;|1-\xi |\le {\daleth }_3\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|((\tau {\left[3\right]}_{q^2}-\beta ){\left[3\right]}_{q^2}{A}_3+T{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_3,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_3=\left|{\displaystyle \frac{((\tau {\left[3\right]}_{q^2}-\beta ){\left[3\right]}_{q^2}{A}_3+T{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-\beta )}^2{\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha (\tau {\left[3\right]}_{q^2}-\beta ){\left[3\right]}_{q^2}{A}_3{(2x-1)}^2}}\right|,$$

and T, $A_2$ and $A_3$ are as mentioned in (16).

As a consequence of Theorem 1, setting $\beta =0$ leads to the following outcome:

Corollary 5.

Let $0\le \gamma \le 1$, and $\tau \ge 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,0,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|(\tau {\left[3\right]}_{q^2}{\gamma }_3{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^2{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{\tau }^2{\left[2\right]}_{q^2}^2{\gamma }_1^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{\tau {\left[3\right]}_{q^2}{\gamma }_3{A}_3}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{\tau {\left[3\right]}_{q^2}{\gamma }_3{A}_3}}\hfill & ;|1-\xi |\le {\daleth }_4\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|(\tau {\left[3\right]}_{q^2}{\gamma }_3{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^2{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_4,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_4=\left|{\displaystyle \frac{(\tau {\left[3\right]}_{q^2}{\gamma }_3{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^2{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha \tau {\left[3\right]}_{q^2}{\gamma }_3{A}_3{(2x-1)}^2}}\right|,$$

$A_2$ and $A_3$ are as stated in (16), and ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ are as stated in (17).

Corollary 5 with $\gamma =0$ becomes:

Corollary 6.

Let $\tau \ge 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,0,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|(\tau {\left[3\right]}_{q^2}{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{\tau }^2{\left[2\right]}_{q^2}^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{\tau {\left[3\right]}_{q^2}{A}_3}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{\tau {\left[3\right]}_{q^2}{A}_3}}\hfill & ;|1-\xi |\le {\daleth }_5\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|(\tau {\left[3\right]}_{q^2}{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_5,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_5=\left|{\displaystyle \frac{(\tau {\left[3\right]}_{q^2}{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha \tau {\left[3\right]}_{q^2}{A}_3{(2x-1)}^2}}\right|,$$

$A_2$ and $A_3$ are as stated in (16).

Corollary 5 with $\gamma =1$ becomes the following:

Corollary 7.

Let $\tau \ge 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,0,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|(\tau {\left[3\right]}_{q^2}^2{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^4{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^4{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{\tau }^2{\left[2\right]}_{q^2}^4{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{\tau {\left[3\right]}_{q^2}^2{A}_3}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{\tau {\left[3\right]}_{q^2}^2{A}_3}}\hfill & ;|1-\xi |\le {\daleth }_6\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|(\tau {\left[3\right]}_{q^2}^2{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^4{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^4{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_6,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_6=\left|{\displaystyle \frac{(\tau {\left[3\right]}_{q^2}^2{A}_3+\frac{\tau (\tau -1)}{2}{\left[2\right]}_{q^2}^4{A}_2^2)\alpha {(2x-1)}^2-2\tau {\left[2\right]}_{q^2}^4{A}_2^2(x^2-x+\frac{1}{6})}{\alpha \tau {\left[3\right]}_{q^2}^2{A}_3{(2x-1)}^2}}\right|,$$

$A_2$ and $A_3$ are as stated in (16).

By applying $\beta =1$, the outcome derived from Theorem 1 is as follows:

Corollary 8.

Let $\tau \ge 1$, and $0\le \gamma \le 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,1,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|((\tau {\left[3\right]}_{q^2}-1){\gamma }_3{A}_3+T_1{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{(\tau {\left[2\right]}_{q^2}-1)}^2{\gamma }_1^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-1){\gamma }_3{A}_3}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-1){\gamma }_3{A}_3}}\hfill & ;|1-\xi |\le {\daleth }_7\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|((\tau {\left[3\right]}_{q^2}-1){\gamma }_3{A}_3+T_1{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_7,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_7=\left|{\displaystyle \frac{((\tau {\left[3\right]}_{q^2}-1){\gamma }_3{A}_3+T_1{\gamma }_2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{\gamma }_1^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha (\tau {\left[3\right]}_{q^2}-1){\gamma }_3{A}_3{(2x-1)}^2}}\right|,$$

$$T_1={\displaystyle \frac{\tau (\tau -1){\left[2\right]}_{q^2}^2}{2}}-\tau {\left[2\right]}_{q^2}+1,$$

(36)

$A_2$ and $A_3$ are as stated in (16), and ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$ are as stated in (17).

The following would result from Corollary 8, if $\gamma =0$:

Corollary 9.

Let $\tau \ge 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,1,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|((\tau {\left[3\right]}_{q^2}-1)A_3+T_1{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{(\tau {\left[2\right]}_{q^2}-1)}^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-1)A_3}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-1)A_3}}\hfill & ;|1-\xi |\le {\daleth }_8\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|((\tau {\left[3\right]}_{q^2}-1)A_3+T_1{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_8,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_8=\left|{\displaystyle \frac{((\tau {\left[3\right]}_{q^2}-1)A_3+T_1{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha (\tau {\left[3\right]}_{q^2}-1)A_3{(2x-1)}^2}}\right|,$$

$T_1$ is as stated in (36), and $A_2$ and $A_3$ are as stated in (16).

Corollary 10.

Let $\tau \ge 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,1,\tau )$, then

$$|d_2|\le {\displaystyle \frac{\left|\alpha (2x-1)\right|\sqrt{|2x-1|}}{\sqrt{2|((\tau {\left[3\right]}_{q^2}-1){\left[3\right]}_{q^2}{A}_3+T_1{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})|}}},$$

$$|d_3|\le {\displaystyle \frac{{\left|\alpha (2x-1)\right|}^2}{{(\tau {\left[2\right]}_{q^2}-1)}^2{\left[2\right]}_{q^2}^2{A}_2^2}}+{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-1){\left[3\right]}_{q^2}{A}_3}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right(2x-1\left)\right|}{(\tau {\left[3\right]}_{q^2}-1){\left[3\right]}_{q^2}{A}_3}}\hfill & ;|1-\xi |\le {\daleth }_7\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|2x-1|}^3|1-\xi |}{2|((\tau {\left[3\right]}_{q^2}-1){\left[3\right]}_{q^2}{A}_3+T_1{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})|}}\hfill & ;|1-\xi |\ge {\daleth }_7,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_7=\left|{\displaystyle \frac{((\tau {\left[3\right]}_{q^2}-1){\left[3\right]}_{q^2}{A}_3+T_1{\left[2\right]}_{q^2}^2{A}_2^2)\alpha {(2x-1)}^2-2{(\tau {\left[2\right]}_{q^2}-1)}^2{\left[2\right]}_{q^2}^2{A}_2^2(x^2-x+\frac{1}{6})}{\alpha (\tau {\left[3\right]}_{q^2}-1){\left[3\right]}_{q^2}{A}_3{(2x-1)}^2}}\right|,$$

$T_1$ is as stated in (36), and $A_2$ and $A_3$ are as stated in (16).

Remark 4.

The classes discussed in Corollaries 2 through 10 are referred to as the τ-pseudo subclasses. By setting $\tau =1$ in each case, one obtains the corresponding results for the classical subclasses: ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,\beta ,1)$, ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,\beta ,1)$, ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,0,1)$, ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,0,1)$, ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,0,1)$, ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,1,1)$, ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,0,1,1)$, and ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,1,1,1)$. Specifically, this reduction yields bounds for the initial coefficients $|d_2|$, $|d_3|$, and for the Fekete–Szegö functional $|d_3-\xi d_2^2|$, where $\xi \in \mathbb{R}$, for functions belonging to the corresponding subclasses considered in Corollaries 2–10.

4. Conclusions

In this work, we have introduced an extensive subclass of regular and bi-univalent functions defined by the $q^2$-Srivastava–Attiya operator in conjunction with Bernoulli polynomials, denoted by ${\mathfrak{T}}_{\sigma ,q^2,b}^s(\alpha ,\gamma ,\beta ,\tau )$. For functions belonging to this family, we established bounds for the Maclaurin coefficients $|d_2|$ and $|d_3|$, and determined the Fekete–Szegö functional $|d_3-\xi d_2^2|$, where $\xi \in \mathbb{R}$. Parameter specialization yields several interesting corollaries, as demonstrated in Section 3. Preliminary observations suggest that this subclass or other subclasses may also be effective when applied to other families of special polynomials, such as Euler polynomials and generalized Bernoulli polynomials. Such an extension would further broaden the scope and applicability of the subclass, enriching both the theoretical framework and its potential mathematical applications. The proposed subclass opens promising avenues for future research, particularly in the analysis of higher-order Hankel and Toeplitz determinants. Most notably, this work bridges the gap between classical and contemporary approaches by demonstrating how the generalized class unifies and extends earlier studies on bi-univalent and subordinate functions. The results presented herein lay a solid foundation for subsequent investigations in this evolving area of mathematical analysis.