Geometric and Functional Symmetries in q-Bernoulli Polynomial Generated Bi-Univalent Function Subfamilies

Abstract

This study is inspired by the rich symmetry and diverse applications of special polynomial families, with a particular focus on the q-Bernoulli polynomials, which have recently emerged as significant tools in bi-univalent function theory. These polynomials are distinguished by their mathematical versatility, analytical manageability, and strong potential for generalization, offering an elegant framework for advancing the study of such functions. In this paper, we introduce a novel subclass of bi-univalent functions defined through q-Bernoulli polynomials. We obtain coefficient estimates for functions in this class and investigate their implications for the Fekete–Szegö functional. Additionally, we present several new results to enrich the theoretical landscape of bi-univalent functions associated with q-Bernoulli polynomials.

1. Preliminaries

Geometric Function Theory (GFT) is a dynamic and influential branch of complex analysis, centered on the symmetry and geometric behavior of analytic functions. It explores how holomorphic functions map domains within the complex plane, focusing on fundamental concepts such as univalent (injective) functions, conformal mappings, and distortion theorems. Rooted in the pioneering work of Riemann, Schwarz, and Koebe, GFT has evolved into a rich field marked by continuous growth. Modern developments in quasiconformal mappings, Teichmüller theory, and connections to mathematical physics further extend its reach. This field’s distinctive blend of geometric intuition and analytical precision imparts it with broad appeal and deep mathematical significance across numerous mathematical disciplines. Within GFT, the study of bi-univalent functions exemplifies this symmetry, enriching the theoretical foundation while opening new directions for practical applications in fluid dynamics, engineering, and conformal mapping theory.

Define the open unit disk by $\mathfrak{U}=$$\{\zeta \in \mathbb{C}:|\zeta |<1\}$. The family 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)

is identified by $\mathcal{A}$. Let $\mathcal{S}$ = $\{\varphi \in \mathcal{A}:\varphi$, is univalent in $\mathfrak{U}$. Bieberbach conjectured that for every $\varphi \in \mathcal{S}$, $|d_j| \le j$, for $j\ge 2$ [1]. Numerous subclasses of $\mathcal{S}$ were developed to investigate this hypothesis, which was ultimately resolved by Branges for all $j\ge 2$ [2]. For functions in $\mathcal{S}$ [3], the Fekete–Szegö functional (FSF) $|d_3-\xi d_2^2|$, $\xi \in \mathbb{R}$, presents another notable problem that has been widely studied in various subclasses. Among these, the class $\sigma$ of bi-univalent functions, introduced by Lewin [4] and defined by $\sigma$ = $\{\varphi \in \mathcal{A}:\varphi$ and $\psi ={\varphi }^{-1}$ are both univalent in $\mathfrak{U}$ is especially prominent. The Koebe theorem (see [5]) asserts that every $\varphi \in \mathcal{S}$ of the form (1) admits an inverse with an expansion of the form

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

(2)

such that $\varsigma =\psi \left(\varphi \right(\varsigma \left)\right),\varsigma \in \mathfrak{U}$, and $w=\varphi \left(\psi \left(w\right)\right),\left|w\right|<r_0\left(\varphi \right),1/4\le r_0\left(\varphi \right),w\in \mathfrak{U}$. Since the functions ${\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\varsigma }{1-\varsigma }}\right)$ and ${\displaystyle \frac{\varsigma }{\sqrt{1-{\varsigma }^2}}}$ are members of the $\sigma$ family, this class is non-empty. ${\displaystyle \frac{2\varsigma -{\varsigma }^2}{2}}$ and the Koebe function ${\displaystyle \frac{\varsigma }{{(1-\varsigma )}^2}}$, despite being in $\mathcal{S}$, are not elements of $\sigma$. For a concise study and insights into several characteristics of the family $\sigma$, refer to [6,7,8,9]. Research on the family of bi-univalent functions have recently gained momentum thanks to Srivastava and his co-authorss for an article [10]. Since this article revived the topic, numerous researchers have looked into a number of fascinating special $\sigma$ families; see [11,12] as well as the references cited therein.

Special polynomials exhibit remarkable symmetrical properties that underscore their crucial role across diverse areas of mathematics and applied sciences. These symmetrical structures contribute to their rich algebraic and analytic behavior, making them indispensable in areas such as combinatorics, computer science, engineering, number theory, numerical analysis, and physics. Their symmetry-driven properties have fueled extensive research in GFT, where such polynomials serve as foundational tools for constructing and analyzing subclasses of analytic functions. In recent developments, particular emphasis has been placed on subclasses of $\mathcal{A}$, especially certain $\sigma$-subfamilies subordinated to classical number sequences or special polynomials. Over the past two decades, substantial attention has been devoted to coefficient problems and functionals such as the FSF $|d_3-\xi d_2^2|$, $\xi \in \mathbb{R}$, for functions within these families (see [13,14,15,16,17,18,19]). These investigations not only enrich the theoretical landscape of GFT but also stimulate applications in mathematical modeling and signal processing.

Bernoulli polynomials possess fundamental significance due to their central role in the Euler–Maclaurin summation formula, which links discrete summation with integral calculus. This classical formula uses Bernoulli numbers and polynomials as key coefficients to approximate sums by integrals, with correction terms expressed via derivatives at boundary points. The detailed analysis by Leinartas and Shishkina [20] elucidates this connection by studying the Euler–Maclaurin formula in the context of summation over lattice points in simplexes, thereby illustrating the natural and essential emergence of Bernoulli polynomials in analytic and combinatorial settings. This foundational importance extends naturally to their q-analogues and provides strong motivation for incorporating q-Bernoulli polynomials into geometric function theory and into the study of subclasses of bi-univalent functions, given their analogous structural and functional roles in these advanced frameworks.

The broad applicability of classical polynomial families has led to numerous extensions and generalizations, among which the q-Bernoulli polynomials stand out for their symmetric structure within the framework of q-calculus. Defined through an appropriate q-generating function, these polynomials form a flexible and robust class that supports the construction of new subclasses of analytic functions. Their symmetry-related features, including a q-Appell-type property, play a crucial role in deriving coefficient bounds and functional inequalities, thereby yielding deeper analytical insight into bi-univalent and related function classes in $\mathfrak{U}$.

We provide an overview of the $\sigma$-derivative operator, a fundamental tool in $\sigma$-calculus with broad applications spanning operator theory, computer science, quantum physics, hypergeometric series, and related fields. We list important definitions and ideas below, presuming that $0<q<1$.

Definition 1. 

Let Φ be a function from the complex plane $\mathbb{C}$ to itself, i.e., Φ:$\mathbb{C}\to \mathbb{C}$. Then the q-derivative of Φ, denoted by $D_q\Phi \left(\varsigma \right)$, is defined by

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

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

This operator generalizes the classical derivative, reducing to it as $q\to 1^{-}$. The $\sigma$-derivative possesses linearity, a product rule adapted to $\sigma$-calculus, and a corresponding quotient and chain rule, making it an analytically versatile operator. Its rich structural properties form the basis for numerous advances across mathematical and physical sciences.

We recall that the q-bracket number, represented by ${\left[j\right]}_q$, is given 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 note that ${\left[j\right]}_q\to j,ifq\to 1^{-}$, and ${\left[0\right]}_q=0$.

For a function $\varphi \in \mathcal{A}$ given by (1), we obtain that $D_q\left(\varphi \left(\varsigma \right)\right)=1+{\displaystyle \sum _{j=2}^{\infty }}{\left[j\right]}_q{d}_j{\varsigma }^{j-1}$. The q-exponential function $e_q$ is defined as $e_q\left(\varsigma \right)={\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{\varsigma }^j}{{\left[j\right]}_q!}}$, $\varsigma \in \mathfrak{U}$. We note that $e\left(\varsigma \right)$ = ${lim}_{q\to 1^{-}}{e}_q\left(\varsigma \right)$ = ${\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{\varsigma }^j}{j!}}$. The q-exponential function $e_q$ is a unique function that satisfies the condition

$${\displaystyle \frac{D_{q}e\left(\varsigma \right)}{D_q\varsigma }}=\sum _{j=0}^{\infty }{\displaystyle \frac{D_q{\varsigma }^j}{{\left[j\right]}_q!}}=\sum _{j=1}^{\infty }{\displaystyle \frac{{\left[j\right]}_q{\varsigma }^{j-1}}{{\left[j\right]}_q!}}=\sum _{j=1}^{\infty }{\displaystyle \frac{{\varsigma }^{j-1}}{{[j-1]}_q!}}=\sum _{j=0}^{\infty }{\displaystyle \frac{{\varsigma }^j}{{\left[j\right]}_q!}}=e_q\left(\varsigma \right),\varsigma \in \mathfrak{U}.$$

The q-derivative operator $d_q$ plays a pivotal role in the analytic investigation of various subclasses of regular functions, underscoring its central importance in modern function theory. The foundational definitions of the q-analogues of the integral and derivative operators, together with some of their applications, were first introduced by Jackson in [21]. Later, Ismail et al. extended this framework by proposing the concept of q-starlike functions in [22]. Following their work, numerous researchers examined q-calculus within the context of geometric function theory. For instance, applications to multivalent functions were discussed in [23]. Using the convolution approach, Zhang et al. [24] studied q-starlike functions associated with generalized conic domains, while Mohammed [25] investigated an operator connected to the q-hypergeometric function, among other related developments.

The q-Bernoulii polynomials [26] ${\mathcal{B}}_{q,j}\left(\varkappa \right)$, with $0<q<1$, $\varkappa \in \mathbb{R}$, and j a non-negative integer, satisfy the following linear homogeneous recurrence relation for $j\ge 2$:

$${\mathcal{B}}_{q,j}\left(\varkappa \right)=q^j\left(\varkappa -{\displaystyle \frac{1}{q{\left[2\right]}_q}}\right){\mathcal{B}}_{q,j-1}\left(\varkappa \right)-{\displaystyle \frac{1}{{\left[j\right]}_q}}\sum _{n=0}^{j-2}{\left(\begin{array}{c}j\\ n\end{array}\right)}_q{q}^{n-1}{b}_{j-n,q}{\mathcal{B}}_{n,q}\left(\varkappa \right).$$

(3)

with initial polynomials

$${\mathcal{B}}_{q,0}\left(\varkappa \right)=1,{\mathcal{B}}_{q,1}\left(\varkappa \right)={\displaystyle \frac{{\left[2\right]}_q\varkappa -q}{{\left[2\right]}_q}}.$$

(4)

and clearly

$${\mathcal{B}}_{q,2}\left(\varkappa \right)=\varkappa (\varkappa -1)+{\displaystyle \frac{q}{{\left[2\right]}_q{\left[3\right]}_q}}.$$

(5)

The generating function of the q-Bernoulii polynomials ${\mathcal{B}}_{q,j}\left(\varkappa \right)$, $\varkappa \in \mathbb{R}$, is given as follows (see [27,28]):

$${\mathcal{B}}_q(\varkappa ,h)={\displaystyle \frac{h{e}_q\left(h\varkappa \right)}{e_q\left(h\right)-1}}=\sum _{j=0}^{\infty }{\mathcal{B}}_{q,j}\left(\varkappa \right){\displaystyle \frac{h^j}{{\left[j\right]}_q!}},\left|h\right|<2\pi .$$

(6)

The choice of q-Bernoulli polynomials in the setting of bi-univalent function theory is driven by their rich symmetric structure and their wide scope for generalization. These polynomials arise from introducing the deformation parameter q, thereby extending the classical Bernoulli polynomials and yielding a flexible analytical framework that aligns naturally with generating function techniques used in coefficient estimation. The associated generating functions fit harmoniously with power series methods that are central to obtaining sharp bounds on initial coefficients, which in turn facilitates the treatment of problems such as the FSF.

The underlying q-calculus structure of these polynomials also forges links between bi-univalent function theory and several other areas, including quantum theory and special functions, thus deepening the theoretical underpinnings and opening avenues for interdisciplinary work. In addition, q-Bernoulli polynomials enable the definition of new q-subclasses with prescribed geometric properties, thereby substantially extending both the range and the depth of current research in GFT.

For ${\mathfrak{a}}_{\mathfrak{1}}$, ${\mathfrak{a}}_{\mathfrak{2}}$$\in \mathcal{A}$ holomorphic in $\mathfrak{U}$, ${\mathfrak{a}}_{\mathfrak{1}}$ is subordinate to ${\mathfrak{a}}_{\mathfrak{2}}$, if there is a Schwarz function $\phi \left(\varsigma \right)$ that is holomorphic in $\mathfrak{U}$ with $\phi \left(0\right)=0$ and $\left|\phi \right(\varsigma \left)\right|<1$, such that ${\mathfrak{a}}_{\mathfrak{1}}\left(\varsigma \right)={\mathfrak{a}}_{\mathfrak{2}}\left(\phi \left(\varsigma \right)\right),\varsigma \in \mathfrak{U}$. This is represented by ${\mathfrak{a}}_{\mathfrak{1}}\prec {\mathfrak{a}}_{\mathfrak{2}}$ or ${\mathfrak{a}}_{\mathfrak{1}}\left(\varsigma \right)\prec {\mathfrak{a}}_{\mathfrak{2}}\left(\varsigma \right)$. In case, if ${\mathfrak{a}}_{\mathfrak{2}}\in \mathcal{S}$, then

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

Building on prior studies of coefficient structures and the FSF in specific subfamilies of $\sigma$ [3], we introduce a comprehensive subclass ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$ of $\sigma$, intrinsically linked to the $\sigma$-Bernoulli polynomials ${\mathcal{B}}_{q,j}\left(\varkappa \right)$ as defined in (3).

Throughout this paper, we utilize the generating function ${\mathcal{B}}_q(\varkappa ,\varsigma )$, where $0<q<1$, $\varkappa \in \mathbb{R}$, j a non-negative integer and $\varsigma \in \mathfrak{U}$, as given explicitly in (6). The inverse function ${\varphi }^{-1}\left(w\right)=\psi \left(w\right)$ for $w\in \mathfrak{U}$, is given by Equation (2) and the complex parameter $\alpha$ is taken to be nonzero unless stated otherwise.

Definition 2. 

Let $\vartheta \ge 1$, $0\le \delta \le 1$, and $0\le \eta \le 1$. A function $\varphi \in \sigma$ is said to belong to the subclass ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$ if the following subordination conditions hold:

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left[{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }\right]}^{\vartheta }}{1-\delta +\delta {\varphi }^{\prime }\left(\varsigma \right)}}\right)+(1-\eta )\left({\displaystyle \frac{\varsigma {\left({\varphi }^{\prime }\left(\varsigma \right)\right)}^{\vartheta }}{(1-\delta )\varsigma +\delta \varphi \left(\varsigma \right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),$$

(7)

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left[{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\vartheta }}{1-\delta +\delta {\psi }^{\prime }\left(w\right)}}\right)+(1-\eta )\left({\displaystyle \frac{w{\left({\psi }^{\prime }\left(w\right)\right)}^{\vartheta }}{(1-\delta )w+\delta \psi \left(w\right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,w).$$

(8)

Remark 1. 

This formulation, introduced for the first time by Swamy and Kala [19], employs a novel linear combination of two expressions whose denominators themselves are linear combinations of analytic components. This innovative structure facilitates the unification of multiple function classes that were previously analyzed in isolation, as demonstrated through Case 1.4 and Case 1.5. By providing a generalized and flexible framework, this approach holds significant promise for advancing the theory of function subclasses within geometric function theory, thereby encouraging further analytical investigations and extensions.

For particular choices of $\vartheta$, $\eta$, and $\delta$, the class ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$ produces the following subfamilies of $\sigma$:

$\mathbf{Case}\mathbf{1}$. If $\vartheta =1$ in the family ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, then we get a subfamily ${\mathfrak{T}}_{\sigma }^1(\alpha ,\eta ,\delta ;q,\varkappa )$ of elements $\varphi \in \sigma$ satisfying

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }}{1-\delta +\delta {\varphi }^{\prime }\left(\varsigma \right)}}\right)+(1-\eta )\left({\displaystyle \frac{\varsigma {\varphi }^{\prime }\left(\varsigma \right)}{(1-\delta )\varsigma +\delta \varphi \left(\varsigma \right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }}{1-\delta +\delta {\psi }^{\prime }\left(w\right)}}\right)+(1-\eta )\left({\displaystyle \frac{w{\psi }^{\prime }\left(w\right)}{(1-\delta )w+\delta \psi \left(w\right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,w),$$

where $0\le \eta \le 1$, and $0\le \delta \le 1$.

$\mathbf{Case}\mathbf{2}$. Allowing $\eta =0$ in the family ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, we get a subfamily ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,\delta ;q,\varkappa )$ of elements $\varphi \in \sigma$ fulfilling

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{\varsigma {\left({\varphi }^{\prime }\left(\varsigma \right)\right)}^{\vartheta }}{(1-\delta )\varsigma +\delta \varphi \left(\varsigma \right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{w{\left({\psi }^{\prime }\left(w\right)\right)}^{\vartheta }}{(1-\delta )w+\delta \psi \left(w\right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,w),$$

where $\vartheta \ge 1$, and $0\le \delta \le 1$.

$\mathbf{Case}\mathbf{3}$. Allowing $\eta =1$ in the family ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, we get a subfamily ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,\delta ;q,\varkappa )$ of elements $\varphi \in \sigma$ satisfying

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{{\left[{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }\right]}^{\vartheta }}{1-\delta +\delta {\varphi }^{\prime }\left(\varsigma \right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\left({\displaystyle \frac{{\left[{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\vartheta }}{1-\delta +\delta {\psi }^{\prime }\left(w\right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,w),$$

where $\vartheta \ge 1$, and $0\le \delta \le 1$.

With $\delta =0$ set in the family ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, we obtain the subfamily ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,0;q,\varkappa )$, which is defined as follows:

$\mathbf{Case}\mathbf{4}$. If a function $\varphi \in \sigma$ satisfies

$$1+{\displaystyle \frac{1}{\alpha }}(\eta {\left[{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }\right]}^{\vartheta }+(1-\eta ){\left({\varphi }^{\prime }\left(\varsigma \right)\right)}^{\vartheta }-1)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}(\eta {\left[{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\vartheta }+(1-\eta ){\left({\psi }^{\prime }\left(w\right)\right)}^{\vartheta }-1)\prec {\mathcal{B}}_q(\varkappa ,w),$$

then $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,0;q,\varkappa )$, where $\vartheta \ge 1$, and $0\le \eta \le 1$.

Similarly, by setting $\delta =1$ in the family ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, we obtain the subfamily ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,1;q,\varkappa )$, which is defined as follows:

$\mathbf{Case}\mathbf{5}$. If a function $\varphi \in \sigma$ satisfies

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left[{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }\right]}^{\nu }}{{\varphi }^{\prime }\left(\varsigma \right)}}\right)+(1-\eta )\left({\displaystyle \frac{\varsigma {\left({\varphi }^{\prime }\left(\varsigma \right)\right)}^{\nu }}{\varphi \left(\varsigma \right)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),$$

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left[{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{{\psi }^{\prime }\left(w\right)}}\right)+(1-\eta )\left({\displaystyle \frac{w{\left({\psi }^{\prime }\left(w\right)\right)}^{n}u}{\left(\psi \right(w)}}\right)-1\right)\prec {\mathcal{B}}_q(\varkappa ,w),$$

then we say that $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,1;q,\varkappa )$, where $\vartheta \ge 1$, and $0\le \eta \le 1$.

The following is the structure of the paper’s content. For members in the class ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, 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 pertinent associations between some of the particular cases and the key observations. We wrap up the study with a few insights in Section 4.

2. Principal Findings

The present section focuses on obtaining coefficient estimates for analytic functions $\varphi$ in the class ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$.

Theorem 1. 

Let $\vartheta \ge 1$, $0\le \eta \le 1$, and $0\le \delta \le 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha (Y+Z){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{X}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

(9)

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{X}^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_{q}Y}},$$

(10)

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_{q}Y}}\hfill & ;|1-\xi |\le \mathsf{Q}\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha (Y+Z){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{X}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge \mathsf{Q},\hfill \end{array}\right.\end{array}$$

(11)

where

$$\mathsf{Q}=\left|{\displaystyle \frac{\alpha (Y+Z){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{X}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^{2}Y}}\right|,$$

(12)

$$X=(1+\eta )(2\vartheta -\delta ),Y=(1+2\eta )(3\vartheta -\delta ),andZ=(1+3\eta )(2\vartheta (\vartheta -1)-2\vartheta \delta +{\delta }^2).$$

(13)

Proof. 

Let $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$. Then, from subordinations (7) and (8), we can write

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left[{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }\right]}^{\vartheta }}{1-\delta +\delta {\varphi }^{\prime }\left(\varsigma \right)}}\right)+(1-\eta )\left({\displaystyle \frac{\varsigma {\left({\varphi }^{\prime }\left(\varsigma \right)\right)}^{\vartheta }}{(1-\delta )\varsigma +\delta \varphi \left(\varsigma \right)}}\right)-1\right)={\mathcal{B}}_q(\varkappa ,\mathfrak{u}\left(\varsigma \right)),$$

(14)

and

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left[{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\vartheta }}{1-\delta +\delta {\psi }^{\prime }\left(w\right)}}\right)+(1-\eta )\left({\displaystyle \frac{w{\left({\psi }^{\prime }\left(w\right)\right)}^{\vartheta }}{(1-\delta )w+\delta \psi \left(w\right)}}\right)-1\right)={\mathcal{B}}_q(\varkappa ,\mathfrak{v}\left(w\right)),$$

(15)

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

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

(16)

Equation (14) can be reformulated in the following manner through basic mathematical steps:

$$1+{\displaystyle \frac{1}{\alpha }}\left(\eta \left({\displaystyle \frac{{\left[{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }\right]}^{\vartheta }}{1-\delta +\delta {\varphi }^{\prime }\left(\varsigma \right)}}\right)+(1-\eta )\left({\displaystyle \frac{\varsigma {\left({\varphi }^{\prime }\left(\varsigma \right)\right)}^{\vartheta }}{(1-\delta )\varsigma +\delta \varphi \left(\varsigma \right)}}\right)-1\right)$$

$$=1+{\displaystyle \frac{1}{\alpha }}X{d}_2\varsigma +{\displaystyle \frac{1}{\alpha }}\left(Y{d}_3+Z{d}_2^2\right){\varsigma }^2+\cdots ,$$

(17)

where X, Y, and Z are as stated in (13), and

$${\mathcal{B}}_q(\varkappa ,\mathfrak{u}\left(\varsigma \right))=1+{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{u}}_1\varsigma +\left[{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{u}}_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathcal{B}}_{q,2}\left(\varkappa \right){\mathfrak{u}}_1^2\right]{\varsigma }^2+\cdots .$$

(18)

By applying a few elementary mathematical steps, Equation (15) can be rewritten as:

$$1+{\displaystyle \frac{1}{\alpha }}\left(\gamma \left({\displaystyle \frac{{\left[{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\tau }}{1-\beta +\beta {\psi }^{\prime }\left(w\right)}}\right)+(1-\gamma )\left({\displaystyle \frac{w{\left({\psi }^{\prime }\left(w\right)\right)}^{\tau }}{(1-\beta )w+\beta \psi \left(w\right)}}\right)-1\right)$$

$$=1-{\displaystyle \frac{1}{\alpha }}X{d}_{2}w+{\displaystyle \frac{1}{\alpha }}\left([Y(2d_2^2-d_3)+Z{d}_2^2]w^2+\cdots \right),$$

(19)

where X, Y, and Z are as stated in (13), and

$${\mathcal{B}}_q(\varkappa ,\mathfrak{v}\left(w\right))=1+{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{v}}_{1}w+\left[{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{v}}_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathcal{B}}_{q,2}\left(\varkappa \right){\mathfrak{v}}_1^2\right]w^2+\cdots .$$

(20)

After comparing the coefficients of like powers in Equations (17) and (18), and using the equality given in (14), the following conclusion is drawn:

$$X{d}_2=\alpha {\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{u}}_1,$$

(21)

$$Y{d}_3+Z{d}_2^2=\alpha [{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{u}}_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathcal{B}}_{q,2}\left(\varkappa \right){\mathfrak{u}}_1^2],$$

(22)

By comparing the coefficients of like powers in Equations (19) and (20), and using the equality given in (15), we arrive at the following conclusion:

$$-X{d}_2=\alpha {\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{v}}_1,$$

(23)

and

$$Y(2d_2^2-d_3)+Z{d}_2^2=\alpha [{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{v}}_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathcal{B}}_{q,2}\left(\varkappa \right){\mathfrak{v}}_1^2].$$

(24)

From (21) and (23), we get

$${\mathfrak{u}}_1=-{\mathfrak{v}}_1,$$

(25)

and

$$2X^2{d}_2^2={\alpha }^2({\mathfrak{u}}_1^2+{\mathfrak{v}}_1^2){\mathcal{B}}_{q,1}^2\left(\varkappa \right).$$

(26)

The sum of Equations (22) and (24) leads to

$$2(Y+Z)d_2^2=\alpha {\mathcal{B}}_{q,1}\left(\varkappa \right)({\mathfrak{u}}_2+{\mathfrak{v}}_2)+{\displaystyle \frac{\alpha }{{\left[2\right]}_q}}{\mathcal{B}}_{q,2}\left(\varkappa \right)({\mathfrak{u}}_1^2+{\mathfrak{v}}_1^2).$$

(27)

Replacing ${\mathfrak{u}}_1^2+{\mathfrak{v}}_1^2$ from Equation (26) into Equation (27), we obtain

$$d_2^2={\displaystyle \frac{{\alpha }^2{\mathcal{B}}_{q,1}^3\left(\varkappa \right)({\mathfrak{u}}_2+{\mathfrak{v}}_2)}{2\left[(Y+Z)\alpha {\mathcal{B}}_{q,1}^2\left(\varkappa \right)-\frac{X^2}{{\left[2\right]}_q}{\mathcal{B}}_{q,2}\left(\varkappa \right)\right]}}.$$

(28)

By applying (16) to ${\mathfrak{u}}_2$,${\mathfrak{v}}_2$, along with using (4) and (5) for ${\mathcal{B}}_{q,1}\left(\varkappa \right)$ and ${\mathcal{B}}_{q,2}\left(\varkappa \right)$, respectively, Equation (9) follows.

The bound on $|d_3|$ is obtained by subtracting Equation (24) from Equation (22). This subtraction yields the following:

$$d_3=d_2^2+{\displaystyle \frac{\alpha {\mathcal{B}}_{q,1}\left(\varkappa \right)({\mathfrak{u}}_2-{\mathfrak{v}}_2)}{2Y}}.$$

(29)

Using the expression for $d_2^2$ from (26) in (29), it follows that

$$|d_3| = {\displaystyle \frac{{\alpha }^2{\mathcal{B}}_{q,1}^2\left(\varkappa \right)({\mathfrak{u}}_1^2+{\mathfrak{v}}_2^2)}{2X^2}}+{\displaystyle \frac{\alpha {\mathcal{B}}_{q,1}\left(\varkappa \right)({\mathfrak{u}}_2-{\mathfrak{v}}_2)}{2Y}}.$$

(30)

We deduce (10) from (30) by applying (4) to ${\mathcal{B}}_{q,1}\left(\varkappa \right)$, along with using (16) for ${\mathfrak{u}}_2$, and ${\mathfrak{v}}_2$.

Finally, we compute the bound on $|d_3-\xi d_2^2|$ using the values of $d_2^2$ and $d_3$ from (28) and (29), respectively. Consequently, we have

$$|d_3-\xi d_2^2| = {\displaystyle \frac{\left|\alpha \right||{\mathcal{B}}_{q,1}\left(\varkappa \right)|}{2}}\left|\left(\mathfrak{B}+{\displaystyle \frac{1}{Y}}\right){\mathfrak{u}}_2+\left(\mathfrak{B}-{\displaystyle \frac{1}{Y}}\right){\mathfrak{v}}_2\right|$$

where

$$\mathfrak{B}={\displaystyle \frac{\alpha (1-\xi ){\mathcal{B}}_{q,1}^2\left(\varkappa \right)}{\alpha (Y+Z){\mathcal{B}}_{q,1}^2\left(\varkappa \right)-\frac{X^2}{{\left[2\right]}_q}{\mathcal{B}}_{q,2}\left(\varkappa \right)}}.$$

Clearly

$$|d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right||{\mathcal{B}}_{q,1}\left(\varkappa \right)|}{Y}}\hfill & ;\left|\mathfrak{B}\right|\le {\displaystyle \frac{1}{Y}}\hfill \\ \left|\alpha \right||{\mathcal{B}}_{q,1}\left(\varkappa \right)\left|\right|\mathfrak{B}|\hfill & ;\left|\mathfrak{B}\right|\ge {\displaystyle \frac{1}{Y}}.\hfill \end{array}\right.$$

(31)

We derive Equation (11) from (31), with $\mathsf{Q}$ as defined in Equation (12). □

Corollary 1. 

If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_{q}Y}}$$

holds, where Y is as defined in (13).

Remark 2. 

Based on the definition described above, various subfamilies of bi-univalent functions governed by q-Bernoulli polynomials can be derived by selecting specific parameters, such as ϑ, η, and δ. We address some of these in the section that follows.

3. Special Cases

By specializing Theorem 1 to the case $\vartheta =1$, we derive the following outcome:

Corollary 2. 

Let $0\le \eta \le 1$, and $0\le \delta \le 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma }^1(\alpha ,\eta ,\delta ;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{|\alpha ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha ((3-\delta ){\eta }_2-\delta (\delta -2){\eta }_3){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2-\delta )}^2{\eta }_1^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{|\alpha ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2-\delta )}^2{\eta }_1^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3-\delta ){\eta }_2}},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3-\delta ){\eta }_2}}\hfill & ;|1-\xi |\le {\mathsf{Q}}_1\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha ((3-\delta ){\eta }_2-\delta (\delta -2){\eta }_3){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2-\delta )}^2{\eta }_1^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_1,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_1=\left|{\displaystyle \frac{\alpha ((3-\delta ){\eta }_2-\delta (\delta -2){\eta }_3){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2-\delta )}^2{\eta }_1^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^2(3-\delta ){\eta }_2}}\right|,$$

$${\eta }_1=1+\eta ,{\eta }_2=1+2\eta ,and{\eta }_3=1+3\eta .$$

(32)

Remark 3. 

If $\varphi \in {\mathfrak{T}}_{\sigma }^1(\alpha ,\eta ,\delta ;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3-\delta ){\eta }_2}}$$

holds, where ${\eta }_2$ is as defined in (32).

Corollary 3. 

Let $\vartheta \ge 1$, and $0\le \delta \le 1$. If $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,\delta ;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{|\alpha ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha ((\vartheta -\delta )(1+2\vartheta )+{\delta }^2){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\vartheta -\delta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{|\alpha ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2\vartheta -\delta )}^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -\delta )}},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -\delta )}}\hfill & ;|1-\xi |\le {\mathsf{Q}}_2\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha ((\vartheta -\delta )(1+2\vartheta )+{\delta }^2){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\vartheta -\delta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_2,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_2=\left|{\displaystyle \frac{\alpha ((\vartheta -\delta )(1+2\vartheta )+{\delta }^2){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\vartheta -\delta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^2(3\vartheta -\delta )}}\right|.$$

Remark 4. 

If $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,\delta ;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -\delta )}}$$

holds.

By specializing Theorem 1 to the case $\eta =1$, we deduce the following outcome:

Corollary 4. 

Let $\vartheta \ge 1$, and $0\le \delta \le 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,\delta ;q,\varkappa )$, then

$$|d_2|\le$$

$${\displaystyle \frac{|\alpha ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha ((\vartheta -\delta )(1+8\vartheta )+4{\delta }^2-2\delta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\vartheta -\delta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{(2\vartheta -\delta )}^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\vartheta -\delta )}},$$

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

$$|d_3-\xi d_2^2|\le$$

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\vartheta -\delta )}}\hfill & ;|1-\xi |\le {\mathsf{Q}}_3\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha ((\vartheta -\delta )(1+8\vartheta )+4{\delta }^2-2\delta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\vartheta -\delta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_3,\hfill \end{array}\right.\end{array}$$

where ${\mathsf{Q}}_3$=

$$\left|{\displaystyle \frac{\alpha ((\vartheta -\delta )(1+8\vartheta )+4{\delta }^2-2\delta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\vartheta -\delta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{3{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^2(3\vartheta -\delta )}}\right|.$$

Remark 5. 

If $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,\delta ;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\vartheta -\delta )}}$$

holds.

Applying $\delta =0$ would result in the following outcome according to Theorem 1:

Corollary 5. 

Let $\vartheta \ge 1$, and $0\le \eta \le 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,0;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha \vartheta (1+2\vartheta +6\vartheta \eta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{\vartheta }^2{(1+\eta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{\vartheta }^2{(1+\eta )}^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\vartheta (1+2\eta )}},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\vartheta (1+2\eta )}}\hfill & ;|1-\xi |\le {\mathsf{Q}}_4\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha \vartheta (1+2\vartheta +6\vartheta \eta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{\vartheta }^2{(1+\eta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_4,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_4=\left|{\displaystyle \frac{\alpha \vartheta (1+2\vartheta +6\vartheta \eta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{\vartheta }^2{(1+\eta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{{\alpha }^2{(3{\left[2\right]}_q\varkappa -q)}^2\vartheta (1+2\eta )}}\right|,$$

Remark 6. 

If $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,0;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\vartheta (1+2\eta )}}$$

holds.

The important special instances from the family ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,0;q,\varkappa )$ corresponding to $\eta =0$ and $\eta =1$ are obtained as follows:

$\mathbf{Instance}\mathbf{1}$. Let $\eta =0$. Then ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,0;q,\varkappa )$, $\nu \ge 1,$ and this represents a family of functions for which $\varphi \in \mathcal{A}$ satisfies the following conditions:

$$1+{\displaystyle \frac{1}{\alpha }}\left({\left(\varphi \left(\varsigma \right)\right)}^{\prime }{)}^{\nu }-1\right)\prec {\mathfrak{B}}_q(x,\varsigma ),and1+{\displaystyle \frac{1}{\alpha }}\left({\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }-1\right)\prec {\mathfrak{B}}_q(x,w).$$

$\mathbf{Instance}\mathbf{2}$. Let $\eta =1$. Then ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,0;q,\varkappa )$, $\nu \ge 1$, and this represents a family of functions for which $\varphi \in \mathcal{A}$ satisfies the following conditions:

$$1+{\displaystyle \frac{1}{\alpha }}\left({\left[{\left(\varsigma {\left(\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }-1\right)\prec {\mathfrak{B}}_q(x,\varsigma ),and1+{\displaystyle \frac{1}{\alpha }}\left({\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }-1\right)\prec {\mathfrak{B}}_q(x,w).$$

The initial coefficient estimates, along with the FSF, for functions belonging to the families ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,0;q,\varkappa )$ and ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,0;q,\varkappa )$ are established in the following two corollaries:

Corollary 6. 

Let $\vartheta \ge 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,0;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha \vartheta (1+2\vartheta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{\vartheta }^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{\vartheta }^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\vartheta }},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\vartheta }}\hfill & ;|1-\xi |\le {\mathsf{Q}}_5\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha \vartheta (1+2\vartheta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{\vartheta }^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_5,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_5=\left|{\displaystyle \frac{\alpha \vartheta (1+2\vartheta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{\vartheta }^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{{\alpha }^2{(3{\left[2\right]}_q\varkappa -q)}^2\vartheta }}\right|,$$

Remark 7. 

If $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,0;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\vartheta }}$$

holds.

Corollary 7. 

Let $\vartheta \ge 1$. If a function $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,0;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha \vartheta (1+8\vartheta ){({\left[2\right]}_q\varkappa -q)}^2-16{\left[2\right]}_q{\vartheta }^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{16{\left[2\right]}_q^2{\vartheta }^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{9{\left[2\right]}_q\vartheta }},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{9{\left[2\right]}_q\vartheta }}\hfill & ;|1-\xi |\le {\mathsf{Q}}_6\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha \vartheta (1+8\vartheta ){({\left[2\right]}_q\varkappa -q)}^2-16{\left[2\right]}_q{\vartheta }^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_6,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_6=\left|{\displaystyle \frac{\alpha \vartheta (1+8\vartheta ){({\left[2\right]}_q\varkappa -q)}^2-16{\left[2\right]}_q{\vartheta }^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{9{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^2\vartheta }}\right|.$$

Remark 8. 

If $\varphi \in {\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,0;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{9{\left[2\right]}_q\vartheta }}$$

holds.

Applying $\delta =1$ would result in the following outcome according to Theorem 1:

Corollary 8. 

Let $\vartheta \ge 1$, and $0\le \eta \le 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,1;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha V{({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\vartheta -1)}^2{(1+\eta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2\vartheta -1)}^2{(1+\eta )}^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -1)(1+2\eta )}},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -1)(1+2\eta )}}\hfill & ;|1-\xi |\le {\mathsf{Q}}_7\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha V{({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\vartheta -1)}^2{(1+\eta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_7,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_7=\left|{\displaystyle \frac{\alpha V{({\left[2\right]}_q\varkappa -q)}^2{-\left[2\right]}_q{(2\vartheta -1)}^2{(1+\eta )}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^2(3\vartheta -1)(1+2\eta )}}\right|,$$

and

$$V=\vartheta (\vartheta -1)(1+6\eta )+\eta +{\vartheta }^2.$$

Remark 9. 

If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,1;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -1)(1+2\eta )}}$$

holds.

The important special instances from the family ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,1;q,\varkappa )$ corresponding to $\eta =0$ and $\eta =1$ are obtained as follows:

$\mathbf{Instance}\mathbf{3}$. Let $\eta =0$. Then ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,1;q,\varkappa )$, $\nu \ge 1,$ and this represents a family of functions for which $\varphi \in \mathcal{A}$ satisfies the following conditions:

$$1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{\varsigma {\left({\varphi }^{\prime }\left(\varsigma \right)\right)}^{\nu }}{\varphi \left(\varsigma \right)}}-1\right)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),and1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{w{\left({\psi }^{\prime }\left(w\right)\right)}^{\nu }}{\psi \left(w\right)}}-1\right)\prec {\mathcal{B}}_q(\varkappa ,w).$$

$\mathbf{Instance}\mathbf{4}$. Let $\eta =1$. Then ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,1;q,\varkappa )$, $\nu \ge 1$, and this represents a family of functions for which $\varphi \in \mathcal{A}$ satisfies the following conditions:

$$1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{{\left[{\left(\varsigma {\varphi }^{\prime }\left(\varsigma \right)\right)}^{\prime }\right]}^{\nu }}{{\varphi }^{\prime }\left(\varsigma \right)}}-1\right)\prec {\mathcal{B}}_q(\varkappa ,\varsigma ),and1+{\displaystyle \frac{1}{\alpha }}\left({\displaystyle \frac{{\left[{\left(w{\psi }^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{{\psi }^{\prime }\left(w\right)}}-1\right)\prec {\mathcal{B}}_q(\varkappa ,w).$$

The initial coefficient bounds, in conjunction with FSF, for functions belonging to the classes ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,1;q,\varkappa )$ and ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,1;q,\varkappa )$ are established in the subsequent two corollaries:

Corollary 9. 

Let $\vartheta \ge 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,1;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha \vartheta (2\vartheta -1){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\vartheta -1)}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2\vartheta -1)}^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -1)}},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -1)}}\hfill & ;|1-\xi |\le {\mathsf{Q}}_8\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha \vartheta (2\vartheta -1){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\vartheta -1)}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_8,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_8=\left|{\displaystyle \frac{\alpha \vartheta (2\vartheta -1){({\left[2\right]}_q\varkappa -q)}^2{-\left[2\right]}_q{(2\vartheta -1)}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^2(3\vartheta -1)}}\right|.$$

Remark 10. 

If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,1;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\vartheta -1)}}$$

holds.

Corollary 10. 

Let $\vartheta \ge 1$. If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,1;q,\varkappa )$, then

$$|d_2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{\sqrt{{\left[2\right]}_q|\alpha (8{\vartheta }^2-7\vartheta +1){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\vartheta -1)}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{(2\vartheta -1)}^2}}+{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\vartheta -1)}},$$

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

$$\begin{array}{c}\hfill |d_3-\xi d_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\vartheta -1)}}\hfill & ;|1-\xi |\le {\mathsf{Q}}_9\hfill \\ {\displaystyle \frac{{\left|\alpha \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\xi |}{{\left[2\right]}_q|\alpha (8{\vartheta }^2-7\vartheta +1){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\vartheta -1)}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-\xi |\ge {\mathsf{Q}}_9,\hfill \end{array}\right.\end{array}$$

where

$${\mathsf{Q}}_9=\left|{\displaystyle \frac{\alpha (8{\vartheta }^2-7\vartheta +1){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\vartheta -1)}^2(\varkappa (\varkappa -1)+\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{3{\alpha }^2{({\left[2\right]}_q\varkappa -q)}^2(3\vartheta -1)}}\right|.$$

Remark 11. 

If $\varphi \in \sigma$ is a member of ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,1;q,\varkappa )$, then under the condition $\xi =1$, the inequality

$$|d_3-d_2^2| \le {\displaystyle \frac{\left|\alpha \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\vartheta -1)}}$$

holds.

Remark 12. 

(i). By setting $q=1$ in Theorem 1, we get explicit bounds for $|d_2|$, $|d_3|$, and $|d_3-\xi d_2^2|$, $\xi \in \mathbb{R}$ for function belonging to the class ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;1,\varkappa )$, which is subordinate to the Bernoulli polynomials.

(ii). Similarly, by setting $q=1$ in Corollaries 1 through 9, we get explicit bounds for the same coefficients and functional for function in the following classes, each subordinate to the Bernoulli polynomials, respectively:

• (i). ${\mathfrak{T}}_{\sigma }^1(\alpha ,\eta ,\delta ;1,\varkappa )$, (ii). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,\delta ;1,\varkappa )$, (iii). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,\delta ;1,\varkappa )$,
• (iv). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,0;1,\varkappa )$, (v). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,0;1,\varkappa )$, (vi). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,0;1,\varkappa )$,
• (vii). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,1;1,\varkappa )$, (viii). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,1;1,\varkappa )$, and (ix). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,1,1;1,\varkappa )$.

Remark 13. 

By setting $\vartheta =1$ in Corollaries 2 through 9 we obtain explicit bounds for $|d_2|$, $|d_3|$, and $|d_3-\xi d_2^2|$, $\xi \in \mathbb{R}$, for function belonging to the following classes, respectively.

• (i). ${\mathfrak{T}}_{\sigma }^1(\alpha ,0,\delta ;1,\varkappa )$, (ii). ${\mathfrak{T}}_{\sigma }^1(\alpha ,1,\delta ;1,\varkappa )$, (iii). ${\mathfrak{T}}_{\sigma }^1(\alpha ,\eta ,0;1,\varkappa )$,
• (iv). ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,0,0;1,\varkappa )$, (v). ${\mathfrak{T}}_{\sigma }^1(\alpha ,1,0;1,\varkappa )$, (vi). ${\mathfrak{T}}_{\sigma }^1(\alpha ,\eta ,1;1,\varkappa )$,
• (vii). ${\mathfrak{T}}_{\sigma }^1(\alpha ,0,1;1,\varkappa )$, and (viii). ${\mathfrak{T}}_{\sigma }^1(\alpha ,1,1;1,\varkappa )$.

4. Conclusions

In this paper, we have established coefficient bounds for functions belonging to a comprehensive class ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;q,\varkappa )$, defined via subordination to the generating function associated with the q-Bernoulli polynomials, which possess inherent symmetry properties. By employing techniques from the theory of differential subordination and leveraging the analytic and symmetric properties of Bernoulli polynomials, we derived estimates for $|d_2|$, $|d_3|$, and the FSF $|d_3-\xi d_2^2|$, where $\xi \in \mathbb{R}$. Several interesting corollaries were obtained by specializing the parameters, illustrating the flexibility and breadth of the defined class.

The analytic formulation of this subclass provides provides a promising symmetry-inspired framework for addressing advanced coefficient problems, including those involving higher-order Hankel and Toeplitz determinants. These determinant-based functionals reveal underlying symmetrical structures with far-reaching implications in geometric function theory, approximation theory, and the study of linear operators. Future investigations may explore determinant inequalities, structural properties, and subclass relationships arising from this family, particularly in connection with other special polynomials and q-calculus formulations.

By setting $q=1$ in Theorem 1, we noted in Remark 12 explicit bounds for $|d_2|$, $|d_3|$, and $|d_3-\xi d_2^2|$, $\xi \in \mathbb{R}$, for functions belonging to the class ${\mathfrak{T}}_{\sigma }^{\vartheta }(\alpha ,\eta ,\delta ;1,\varkappa )$, which is subordinate to the Bernoulli polynomials, known for their classical symmetry properties. Similarly, by applying the condition $q=1$ in Corollaries 1 through 9, explicit bounds for the same coefficients and the FSF were obtained for functions in the respective subclasses, each subordinate to the Bernoulli polynomials, as noted in Remark 12.