Certain Subclasses of Te-Univalent Functions Subordinate to q-Bernoulli Polynomials

Abstract

The present work centers on the significance of q-calculus in geometric function theory and its expanding applications within the domain of Te-univalent functions, especially those associated with special polynomials like the q-Bernoulli polynomials. Motivated by recent interest in these polynomials, our study introduces and analyzes a generalized subclass of Te-univalent functions that intimately relate to q-Bernoulli polynomials. For this new family, we establish explicit bounds for $|d_2|$ and $|d_3|$, and provide estimates for the Fekete–Szegö functional $|d_3-\xi d_2^2|, \xi \in \mathbb{R}$. Our findings contribute new results and demonstrate meaningful connections to prior work involving Te-univalent and subordinate functions, thereby broadening and integrating various strands of the existing literature.

1. Preliminaries

Geometric Function Theory (GFT) is a vibrant and evolving area of complex analysis that continues to draw significant research interest. A key focus within GFT is the study of holomorphic univalent functions and their diverse subclasses. In recent years, bi-univalent function classes have gained considerable attention due to their intricate geometric properties and complex coefficient problems. More recently, Te-univalent function classes have been introduced, with initial studies addressing their coefficient estimates. These Te-univalent classes provide fresh opportunities to explore functional inequalities, refine coefficient bounds, and develop operator-based generalizations. This work contributes to the growing literature by introducing and analyzing new subfamilies of Te-univalent functions connected to q–Bernoulli polynomials.

Let $\mathfrak{U}=$$\{\varsigma \in \mathbb{C}:|\varsigma |<1\}$. Consider the class $\mathcal{A}$ 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)

and define $\mathcal{S}$ as the set of functions $\varphi$ from $\mathcal{A}$ that are one-to-one (univalent) in $\mathfrak{U}$. In [1], Bieberbach conjectured that for every function $\varphi \in \mathcal{S}$, the coefficients satisfy $|d_j|\le j,j\ge 2$. In the effort to prove this conjecture, many new subclasses of $\mathcal{S}$ were introduced, and numerous significant results were obtained. Researchers devoted considerable effort to this problem over several decades, until it was ultimately resolved by Louis de Branges, who proved the conjecture for all $j\ge 2$ in [2]. Another central problem in GFT is the Fekete–Szegö functional (FSF), given by $|d_3-\xi d_2^2|, \xi \in \mathbb{R}$, which is studied for functions $\varphi \in \mathcal{S}$ [3]. There’s been a bunch of research around this, especially on functions in various subclasses of $\mathcal{S}$. One notable group among them is the bi-univalent functions, labeled $\sigma$. This concept was introduced by Lewin in [4]. A function $\varphi$ belongs to $\sigma$ if both $\varphi$ and its inverse $\psi$ = ${\varphi }^{-1}$ are univalent in the unit disk $\mathfrak{U}$; that is, $\sigma$=$\{\varphi \in \mathcal{A}:\varphi$ and $\psi$ = ${\varphi }^{-1}$ are both univalent in $\mathfrak{U}$}. According to the well-known Koebe theorem (see [5]), every function $\varphi \in \mathcal{S}$ has an inverse, which is given by

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

(2)

satisfying $\varsigma =\psi \left(\varphi \right(\varsigma \left)\right)$ 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 several functions that belong to it, such as ${\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\varsigma }{1-\varsigma }}\right),-log(1-\varsigma )$ and ${\displaystyle \frac{\varsigma }{1-\varsigma }}$. However, functions like $\varsigma -{\displaystyle \frac{{\varsigma }^2}{2}}$, ${\displaystyle \frac{\varsigma }{1-{\varsigma }^2}}$, and the Koebe function ${\displaystyle \frac{\varsigma }{{(1-\varsigma )}^2}}$ although members of the class $\mathcal{S}$, do not belong to $\sigma$. For a concise overview and insight into some properties of the class $\sigma$, see [6,7,8,9]. A renewed interest in the study of this class was sparked by the influential paper by Srivastava and his collaborators [10]. Following this work, numerous researchers have explored various intriguing subclasses of $\sigma$; see [11,12,13,14,15] and the references therein.

For the function $\varphi \in \mathcal{A}$, define the operator $\mathcal{T}$: $\mathcal{A}\to \mathcal{A}$ by

$$\mathcal{T}\varphi \left(\varsigma \right)=\varsigma +t_2{d}_2{\varsigma }^2+t_3{d}_3{\varsigma }^3+\cdots =\varsigma +\sum _{j=2}^{\infty }{t}_j{d}_j{\varsigma }^j,t_j\in \mathbb{C},and\varsigma \in \mathfrak{U}.$$

(3)

Let ${\mathcal{T}}_{\mathcal{S}}$ denote the family of all univalent functions in $\mathfrak{U}$ that are represented by (3). Every function $\mathcal{T}\varphi \in {\mathcal{T}}_{\mathcal{S}}$ has an inverse ${\left(\mathcal{T}\varphi \right)}^{-1}$, given by ${\left(\mathcal{T}\varphi \right)}^{-1}\left(\mathcal{T}\varphi \left(\varsigma \right)\right)=\varsigma$, and $\mathcal{T}\varphi \left({\left(\mathcal{T}\varphi \right)}^{-1}\left(w\right)\right)=w$, $\left|w\right|<r_0\left(\mathcal{T}\varphi \right);1/4\le r_0\left(\mathcal{T}\varphi \right),\varsigma ,w\in \mathfrak{U}$, where

$${\left(\mathcal{T}\varphi \right)}^{-1}\left(w\right)=w-t_2{d}_2{w}^2+(2t_2^2{d}_2^2-t_3{d}_3)w^3-(5t_2^3{d}_2^3-5t_2{t}_3{d}_2{d}_3+t_4{d}_4)w^4+\cdots .$$

(4)

If both $\mathcal{T}\varphi$ and ${\left(\mathcal{T}\varphi \right)}^{-1}$ are univalent in $\mathfrak{U}$, then the function $\varphi$ given by (1) is said to be Te-univalent in $\mathfrak{U}$ associated with the operator $\mathcal{T}$. Let ${\mathcal{T}}_{\sigma }$ denote the class of all Te-univalent functions in $\mathfrak{U}$ associated with $\mathcal{T}$, and are represented by (1). When $\mathcal{T}\varphi =\varphi$, we have ${\mathcal{T}}_{\sigma }=\sigma$ and if $t_j\ne 1$ for some j, we have $\mathcal{T}\varphi \left(\mathcal{T}\psi \left(w\right)\right)=w+2[t_3-t_2^2]d_2^2{w}^3+\cdots \ne w$, where $\psi$ is given by (2).

Example illustrating non-emptiness of ${\mathcal{T}}_{\sigma }$:

Consider the normalized Koebe function $k\left(\varsigma \right)={\displaystyle \frac{\varsigma }{{(1-\varsigma )}^2}}=\varsigma +2{\varsigma }^2+3{\varsigma }^3+\cdots$, which is a classical univalent function in $\mathfrak{U}$. For the operator $\mathcal{T}$ defined by (3), if the coefficients $t_j$ are chosen sufficiently close to 1, then $\mathcal{T}k\left(\varsigma \right)$ is a small perturbation of $k\left(\varsigma \right)$. By the stability of univalence under small perturbations and standard distortion theorems, $\mathcal{T}k\left(\varsigma \right)$ remains univalent in $\mathfrak{U}$. Moreover, since the inverse function of the Koebe function is known explicitly and univalent on a certain disk, and $\mathcal{T}$ acts linearly on coefficients, the inverse ${\left(\mathcal{T}k\left(\varsigma \right)\right)}^{-1}$ also remains univalent in a sufficiently small radius around zero, ensuring $\mathcal{T}k\left(\varsigma \right)\in {\mathcal{T}}_{\sigma }$.

The convolution (or Hadamard product) of the function $\varphi$ defined by (1) and the function $\Omega$ given by $\Omega \left(\varsigma \right)=\varsigma +c_1\varsigma +c_2{\varsigma }^2+\cdots =\varsigma +{\displaystyle \sum _{j=2}^{\infty }}{c}_j{\varsigma }^j$ is defined by $(\varphi \ast \Omega )\left(\varsigma \right)=\varsigma +{\displaystyle \sum _{j=2}^{\infty }}{d}_j{c}_j{\varsigma }^j=(\Omega \ast \varphi )\left(\varsigma \right)$ (see [16,17]).

For complex parameters ${\eta }_1,{\eta }_2,\cdots ,{\eta }_p$ and ${\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_s,({\kappa }_n\ne 0,-1,-2,\cdots ,n=1,2,3,\cdots ,s)$, the generalized hypergeometric ${\mathrm{function} }_p{F}_s({\eta }_1,{\eta }_2,\cdots ,{\eta }_p;{\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_s,\varsigma )={\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \frac{{\left({\eta }_1\right)}_j,\cdots ,{\left({\eta }_p\right)}_j}{{\left({\kappa }_1\right)}_j,\cdots ,{\left({\kappa }_s\right)}_j}}{\displaystyle \frac{{\varsigma }^j}{j!}},$ where $p\le s+1,p,s\in {\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\},\varsigma \in \mathfrak{U},$ and ${(⅁)}_j$ is the shift factorial (or Pochhammer symbol) defined in terms of the Gamma function $\Gamma$, by

$$\begin{array}{c}\hfill {(⅁)}_j={\displaystyle \frac{\Gamma (⅁+j)}{\Gamma (⅁)}}=\left\{\begin{array}{cc}1\hfill & ;j=0\hfill \\ ⅁(⅁+1)\cdots (⅁+j-1)\hfill & ;j\in \mathbb{N}.\hfill \end{array}\right.\end{array}$$

Corresponding to the function $\mathfrak{H}({\eta }_1,\cdots ,{\eta }_p;{\kappa }_1,\cdots ,{\kappa }_s;\varsigma )$ defined by

$$\mathfrak{H}({\eta }_1,\cdots ,{\eta }_p;{\kappa }_1,\cdots ,{\kappa }_s;\varsigma )={\varsigma }_p{F}_s({\eta }_1,{\eta }_2,\cdots ,{\eta }_p;{\kappa }_1,{\kappa }_2,\cdots ,{\kappa }_s;\varsigma ),\varsigma \in \mathfrak{U},$$

Dziok and Srivastava [18] introduced the operator

$$\mathcal{H}({\eta }_1,\cdots ,{\eta }_p;{\kappa }_1,\cdots ,{\kappa }_s):\mathcal{A}\to \mathcal{A}$$

defined via the convolution (or Hadamard product) as follows:

$$\mathcal{H}({\eta }_1,\cdots ,{\eta }_p;{\kappa }_1,\cdots ,{\kappa }_s)\varphi \left(\varsigma \right)=\mathfrak{H}({\eta }_1,\cdots ,{\eta }_p;{\kappa }_1,\cdots ,{\kappa }_s;\varsigma )*\varphi \left(\varsigma \right),$$

where $p\le s+1,q,s\in {\mathbb{N}}_0,\varsigma \in \mathfrak{U}$.

If $\varphi \in \mathcal{A}$ is given by (1), then we have

$$\mathcal{H}({\eta }_1,\cdots ,{\eta }_p;{\kappa }_1,\cdots ,{\kappa }_s)\varphi \left(\varsigma \right)=\varsigma +\sum _{j=2}^{\infty }{\Gamma }_j[{\eta }_1;{\kappa }_1]d_j{\varsigma }^j,\varsigma \in \mathfrak{U},$$

(5)

where

$${\Gamma }_j[{\eta }_1;{\kappa }_1]={\displaystyle \frac{{\left({\eta }_1\right)}_{j-1}\cdots {\left({\eta }_p\right)}_{j-1}}{{\left({\kappa }_1\right)}_{j-1}\cdots {\left({\kappa }_s\right)}_{j-1}}}{\displaystyle \frac{1}{(j-1)!}}.$$

(6)

For simplicity of notation, we denote $\mathcal{H}({\eta }_1,\cdots ,{\eta }_p;{\kappa }_1,\cdots ,{\kappa }_s)\varphi ={\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi$. Let ${\mathcal{T}}_{\mathcal{S}}^{p,s}[{\eta }_1;{\kappa }_1]$ denote the family of all functions represented by (5) that are univalent in $\mathfrak{U}$. Every function ${\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \in {\mathcal{T}}_{\mathcal{S}}^{p,s}[{\eta }_1;{\kappa }_1]$ has an inverse ${\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \right)}^{-1}$, defined by $\mathcal{G}\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)=\varsigma$, and ${\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\mathcal{G}\left(w\right)\right)=w$, $\left|w\right|<r_0\left({\mathcal{H}}_{q,s}[{\eta }_1;{\kappa }_1]\varphi \right);1/4\le r_0\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \right),\varsigma ,w\in \mathfrak{U}$, where

$$\mathcal{G}\left(w\right)={\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \right)}^{-1}\left(w\right)=w-{\Gamma }_2[{\eta }_1;{\kappa }_1]d_2{w}^2+[2{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2-{\Gamma }_3[{\eta }_1;{\kappa }_1]d_3]w^3-\cdots .$$

(7)

A function $\varphi \in \mathcal{A}$ is said to be Te-univalent in $\mathfrak{U}$ associated with the operator ${\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]$, if both ${\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi$ and ${\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \right)}^{-1}$ are univalent in $\mathfrak{U}$. Let ${\mathcal{T}}_{\sigma }^{p,s}[{\eta }_1;{\kappa }_1]$ denote the family of all functions $\varphi \in \mathcal{A}$, which are Te-univalent in $\mathfrak{U}$ associated with ${\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]$.

Remark 1.  (i). For $p=2,s=1,{\eta }_1={\kappa }_1=c,$ and ${\eta }_2=1,$ we have ${\mathcal{T}}_{\sigma }^{2,1}[c,1;c]=\sigma .$

(ii). If ${\Gamma }_j{[{\eta }_1;{\kappa }_1]}_j\ne 1$ for some j, we have ${\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\psi \left(w\right)\right)=w+2[{\Gamma }_3[{\eta }_1;{\kappa }_1]-{\Gamma }_2^2[{\eta }_1;{\kappa }_1]]d_2^2{w}^3+\cdots \ne w$, where ψ is given by (2).

Polynomials like Bernoulli, Fibonacci, bivariate Fibonacci, Gegenbauer, Horadam, Lucas-Balancing, and Lucas-Lehmer, along with their various generalizations, hold a fundamental place across numerous fields in applied sciences and mathematics. These areas include combinatorics, computer science, engineering, number theory, numerical analysis, and physics. Their rich algebraic and analytic features have sparked extensive research, especially within GFT, where they help in constructing and studying subclasses of regular functions.

Given their extensive applications, numerous extensions and generalizations of these polynomial families have emerged in the literature. Among them, q-Bernoulli polynomials, a q-calculus-based generalization of classical Bernoulli polynomials, have gained notable attention recently (see [19]). Developed through the framework of q-calculus [20], these polynomials offer a versatile foundation for defining new classes of analytic functions. Their structural qualities enable the derivation of sharper coefficient bounds and functional inequalities, providing deeper insights into bi-univalent functions and related classes within the unit disk.

Within GFT, recent work has increasingly targeted subclasses of regular functions, particularly those in specific $\sigma$-subclasses, that are subordinate to famous polynomial sequences. Over the past several decades, many authors have explored coefficient problems and functionals such as the FSF, for elements of these families (see [21,22,23,24,25,26]). These studies not only contribute to the theoretical framework of GFT but also open avenues for practical applications across mathematical modeling and signal processing.

A deeper mathematical motivation for employing Bernoulli polynomials emerges from their pivotal role in summation theory, exemplified by the Euler-Maclaurin summation formula. This classical formula establishes a vital connection between discrete summation and integral calculus, utilizing Bernoulli numbers and polynomials as fundamental coefficients. The comprehensive study by Leinartas and Shishkina [27] further elucidates this point by analyzing the Euler-Maclaurin formula in the context of summation over lattice points within simplexes, thereby revealing the natural and fundamental appearance of Bernoulli polynomials in analytic and combinatorial problems. This important foundational role extends to their q-analogues, providing strong motivation for integrating q-Bernoulli polynomials into the framework of geometric function theory and the study of subclasses of Te-univalent functions.

Here, we review the q-derivative operator, a basic q-calculus tool that is important in many domains, including operator theory, computer science, quantum physics, hypergeometric series, and other related disciplines. 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.

The q-bracket number, denoted by ${\left[j\right]}_q$, is given by ${\left[j\right]}_q=1+q+q^2+\cdots +q^{j-1}={\displaystyle \frac{1-q^j}{1-q}},(q\ne 1)$. 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 ${\left[0\right]}_q=0$. So, $D_q\Phi \left(\varsigma \right)\to {\Phi }^{\prime }\left(\varsigma \right)$ as $q\to 1^{-}$.

For a function $\varphi \in \mathcal{A}$ defined by (1), we derive 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 $D_q$ serves as a cornerstone in analyzing various subfamilies of holomorphic functions, underscoring its strong connection with GFT. The notion of q-analogues of integral and derivative operators, along with their early applications, was first introduced by Jackson in [20], laying the foundation for modern q-calculus. Building upon Jackson’s work, Ismail et al. extended the framework by proposing a q-analogue of the class of q-starlike functions in [28]. Later, the framework was broadened through the introduction of fractional q-calculus operators, as presented in [29].

These developments spurred extensive research exploring the applications of q-calculus in GFT. For instance, the authors of [29] defined the q-analogue of the Ruscheweyh differential operator, thereby expanding the analytical toolkit available for studying geometric function classes. In [30,31], the q-calculus framework was applied to multivalent functions, while Zhang et al. [32] investigated q-starlike functions associated with generalized conic domains using the convolution (Hadamard product) approach. Furthermore, the work in [33] explored an operator constructed via the q-hypergeometric function, deepening insights into the interplay between q-calculus and analytic function theory.

Such advancements demonstrate how the q-derivative operator bridges classical and modern analytical structures, providing a versatile foundation for defining novel subclasses, establishing coefficient estimates, and formulating functional inequalities within the unit disk.

The q-Bernoulii polynomials (see [34]) ${\mathcal{B}}_{q,j}\left(\varkappa \right)$, with $0<q<1$, $\varkappa \in \mathbb{R}$, and j a non-negative integer, satisfy the following linear homogeneous recurrence relation for $j\ge 2$:

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

(8)

with initial polynomials

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

(9)

and clearly

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

(10)

The generating function of the q-Bernoulii polynomials ${\mathcal{B}}_{q,j}\left(\varkappa \right)$, $\varkappa \in \mathbb{R}$, is given as follows (see [19,35]):

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

(11)

The introduction of the deformation parameter q through q-Bernoulli polynomials offers a powerful and flexible framework for extending classical results in GFT. By enabling interpolation between well-established function classes and facilitating the construction of novel ones, these polynomials play a central role in the study of subordination, superordination, and operator-related problems. Their importance stems from their ability to define q-analogues of differential and integral operators, which underpin key analytic techniques in GFT. The rich analytic structure and intrinsic connections of q-Bernoulli polynomials to special functions, together with the adjustable parameter q, allow for sharper coefficient estimates, refined growth bounds, and enhanced geometric characterizations of analytic functions. Moreover, they serve as a conceptual bridge linking classical analysis to emerging areas such as quantum calculus and mathematical physics, thereby broadening the scope and impact of univalent function theory.

For ${\phi }_1$, ${\phi }_2$$\in \mathcal{A}$ both analytic in $\mathfrak{U}$, the function ${\phi }_1$ is said to be subordinate to ${\phi }_2$ if there is a Schwarz function $\theta \left(\varsigma \right)$, analytic in $\mathfrak{U}$, such that $\theta \left(0\right)=0$ and $\left|\theta \right(\varsigma \left)\right|<1$, with the relation ${\phi }_1\left(\varsigma \right)={\phi }_2\left(\theta \left(\varsigma \right)\right),\varsigma \in \mathfrak{U}$. This subordination is denoted by ${\phi }_1\prec {\phi }_2$ or ${\phi }_1\left(\varsigma \right)\prec {\phi }_2\left(\varsigma \right)$ $(\varsigma \in \mathfrak{U})$. In the special case where ${\phi }_2\in \mathcal{S}$, we have

$${\phi }_1\left(\varsigma \right)\prec {\phi }_2\left(\varsigma \right)\iff {\phi }_1\left(0\right)={\phi }_2\left(0\right)and{\phi }_1\left(\mathfrak{U}\right)\subset {\phi }_2\left(\mathfrak{U}\right).$$

In recent years, the study of Te-univalent function classes has witnessed significant advancements, with particular emphasis on generalized subclasses characterized by differential operators and subordination principles [36,37]. Among these, the class ${\mathcal{T}}_{\sigma }^{p,s}[{\eta }_1;{\kappa }_1]$ has attracted considerable attention due to its remarkable ability to unify and extend numerous known subclasses through its parametric flexibility. Notably, recent investigations by Saravanan [38] and Swamy et al. [39] have focused on structured subclasses of ${\mathcal{T}}_{\sigma }^{p,s}[{\eta }_1;{\kappa }_1]$ that are subordinate to specific polynomial sequences. This subordination approach substantially enriches the theory of analytic functions by providing sharper coefficient estimates, refined growth theorems, and precise distortion bounds. Such studies delve deeply into the geometric properties of these functions and open promising new directions for the analytical characterization of complex-valued functions.

Motivated by these developments, the present work introduces a novel subclass of ${\mathcal{T}}_{\sigma }^{p,s}[{\eta }_1;{\kappa }_1]$, denoted by ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,x)$, consisting of functions subordinate to the q-Bernoulli polynomials. For this subclass, we establish rigorous bounds on the initial coefficients and investigate the Fekete–Szegö functional, advancing the existing theoretical framework. Through these contributions, this paper not only extends the landscape of Te-univalent function theory but also furnishes new analytical tools that leverage the intrinsic algebraic structure of q-Bernoulli polynomials.

Throughout this paper, we employ the GF ${\mathcal{B}}_q(\varkappa ,\varsigma )$, with $0<q<1$, $\varkappa \in \mathbb{R}$, j a non-negative integer and $\varsigma \in \mathfrak{U}$, as defined in (11) and the function $\mathcal{G}\left(w\right)={\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \right)}^{-1}\left(w\right)$ as defined in (7). Unless otherwise specified, the parameter $\rho$ is assumed to lie in $\mathbb{C}\setminus \left\{0\right\}$, and the indices satisfy $p\le s=1,p,s\in {\mathbb{N}}_0$.

Definition 2.

Let $\varkappa \in \mathbb{R}$, $0\le \tau \le 1$, $\nu \ge 1$, and $0\le \delta \le 1$. If $\varphi \in \mathcal{A}$ satisfies

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }}{1-\delta +\delta {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }}}\right)+(1-\tau )\left({\displaystyle \frac{\varsigma {\left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\nu }}{(1-\delta )\varsigma +\delta {\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)}}\right)-1\right)$$

$$\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),\varsigma \in \mathfrak{U},$$

(12)

and

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{1-\delta +\delta {\mathcal{G}}^{\prime }\left(w\right)}}\right)+(1-\tau )\left({\displaystyle \frac{w{\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }}{(1-\delta )w+\delta \mathcal{G}\left(w\right)}}\right)-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),w\in \mathfrak{U},$$

(13)

then we say that $\varphi \in {\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$.

Remark 2.

This definition introduces, for the first time in Te-univalent functions theory, a formulation involving a linear combination of two expressions where each term’s denominator is itself a linear combination of two distinct expressions. This innovative construction enables the blending of function classes that were traditionally studied independently, as exemplified in Cases 4 and 5. By unifying these previously separate classes within a single framework, this approach opens novel perspectives and is poised to stimulate renewed interest and further exploration among researchers in GFT. The resulting interplay enriches the analytical structure and broadens the scope for developing new subclasses and deriving more comprehensive coefficient estimates and geometric properties.

For particular choices of $\tau$, $\delta$, and $\nu$, the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$ produces the following subfamilies of $\sigma$:

1. Let $\tau =0$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,\delta ,\varkappa )$, $\varkappa \in \mathbb{R}$, $\nu \ge 1$, and $0\le \delta \le 1$, is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left(\left({\displaystyle \frac{\varsigma {\left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\nu }}{(1-\delta )\varsigma +\delta {\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)}}\right)-1\right)\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),\varsigma \in \mathfrak{U},$$

and

$$1+{\displaystyle \frac{1}{\varrho }}\left(\left({\displaystyle \frac{w{\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }}{(1-\delta )w+\delta \mathcal{G}\left(w\right)}}\right)-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),w\in \mathfrak{U}.$$

2. Let $\tau =1$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,\delta ,\varkappa )$, $\varkappa \in \mathbb{R}$, $\nu \ge 1$, and $0\le \delta \le 1$, represents the set of functions $\varphi \in \mathcal{A}$ that meet

$$1+{\displaystyle \frac{1}{\varrho }}\left(\left({\displaystyle \frac{{\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }}{1-\delta +\delta {\left({\mathcal{H}}_{q,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }}}\right)-1\right)\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),\varsigma \in \mathfrak{U},$$

and

$$1+{\displaystyle \frac{1}{\varrho }}\left(\left({\displaystyle \frac{{\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{1-\delta +\delta {\mathcal{G}}^{\prime }\left(w\right)}}\right)-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),w\in \mathfrak{U}.$$

3. Let $\nu =1$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,1,\delta ,\varkappa )$, $\varkappa \in \mathbb{R}$, $0\le \tau \le 1$, and $0\le \delta \le 1$, is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}{1-\delta +\delta {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }}}\right)+(1-\tau )\left({\displaystyle \frac{\varsigma \left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}{(1-\delta )+\delta {\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)}}\right)-1\right)$$

$$\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),\varsigma \in \mathfrak{U},$$

and

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}{1-\delta +\delta {\mathcal{G}}^{\prime }\left(w\right)}}\right)+(1-\tau )\left({\displaystyle \frac{w\left({\mathcal{G}}^{\prime }\left(w\right)\right)}{(1-\delta )+\delta \mathcal{G}\left(w\right)}}\right)-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),w\in \mathfrak{U}.$$

4. Let $\delta =0$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,0,\varkappa )$, $\varkappa \in \mathbb{R}$, $0\le \tau \le 1$, and $\nu \ge 1,$ is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau {\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }+(1-\tau ){\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }{)}^{\nu }-1\right)\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),\varsigma \in \mathfrak{U},$$

and

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau {\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }+(1-\tau ){\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),w\in \mathfrak{U}.$$

5. Let $\delta =1$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,1,\varkappa )$, $\varkappa \in \mathbb{R}$, $0\le \tau \le 1$, and $\nu \ge 1,$ is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }}{{\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }}}\right)+(1-\tau )\left({\displaystyle \frac{\varsigma {\left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\nu }}{{\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)}}\right)-1\right)$$

$$\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),\varsigma \in \mathfrak{U},$$

and

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{{\mathcal{G}}^{\prime }\left(w\right)}}\right)+(1-\tau )\left({\displaystyle \frac{w{\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }}{\mathcal{G}\left(w\right)}}\right)-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),w\in \mathfrak{U}.$$

In Section 2, we derive estimates for the coefficients $|d_2|$, $|d_3|$, and the functional $|d_3-\xi d_2^2|,\xi \in \mathbb{R}$ for functions in the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$. In Section 3, interesting outcomes from the results obtained for the defined class are presented, highlighting their connections to previously published findings.

2. Key Findings

In this section, we establish coefficient bounds for arbitrary functions $\varphi$ belonging to the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$.

Theorem 1.

Let $\varkappa ,\xi \in \mathbb{R}$, $0\le \tau \le 1$, $\nu \ge 1$, and $0\le \tau \le 1$. If $\varphi \in \mathcal{A}$ is an element of the family ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\Gamma }_2[{\eta }_1;{\kappa }_1]}}\sqrt{{\displaystyle \frac{{|\left[2\right]}_q\varkappa -q|}{{\left[2\right]}_q|\varrho (\mathsf{M}+\mathsf{N}){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{\mathsf{L}}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

(14)

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{\mathsf{L}}^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

(15)

and

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

(16)

where

$$\daleth =\left|{\displaystyle \frac{\varrho (\mathsf{M}+\mathsf{N}){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{\mathsf{L}}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{\varrho \mathsf{M}{({\left[2\right]}_q\varkappa -q)}^2}}\right|,$$

(17)

$$\mathsf{L}=(\tau +1)(2\nu -\delta ),\mathsf{M}=(2\tau +1)(3\nu -\delta ),and\mathsf{N}=(2\nu (\nu -1)-2\nu \delta +{\delta }^2)(3\tau +1).$$

(18)

Proof. 

Let $\varphi \in {\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$. Then, from (12) and (13), it follows that

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }}{1-\delta +\delta {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }}}\right)+(1-\tau )\left({\displaystyle \frac{\varsigma {\left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\nu }}{(1-\delta )\varsigma +\delta {\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)}}\right)-1\right)$$

$$={\mathfrak{B}}_q(\varkappa ,\mathfrak{l}\left(\varsigma \right)),$$

(19)

and

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{1-\delta +\delta {\mathcal{G}}^{\prime }\left(w\right)}}\right)+(1-\tau )\left({\displaystyle \frac{w{\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }}{(1-\delta )w+\delta \mathcal{G}\left(w\right)}}\right)-1\right)={\mathfrak{B}}_q(\varkappa ,\mathfrak{m}\left(w\right)),$$

(20)

where

$$\mathfrak{l}\left(\varsigma \right)=\sum _{j=1}^{\infty }{\mathfrak{l}}_j{\varsigma }^j={\mathfrak{l}}_1\varsigma +{\mathfrak{l}}_2{\varsigma }^2+{\mathfrak{l}}_3{\varsigma }^3+\cdots ,\varsigma \in \mathfrak{U},$$

and

$$\mathfrak{m}\left(w\right)=\sum _{j=1}^{\infty }{\mathfrak{m}}_j{w}^j={\mathfrak{m}}_{1}w+{\mathfrak{m}}_2{w}^2+{\mathfrak{m}}_3{w}^3+\cdots ,w\in \mathfrak{U},$$

are Schwarz functions with the property (see [5])

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

(21)

By applying a few basic mathematical steps, we can rewrite Equation (19) as

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }}{1-\delta +\delta {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }}}\right)+(1-\tau )\left({\displaystyle \frac{\varsigma {\left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\nu }}{(1-\delta )\varsigma +\delta {\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)}}\right)-1\right)=$$

$$1+{\displaystyle \frac{\mathsf{L}}{\varrho }}{\Gamma }_2[{\eta }_1;{\kappa }_1]d_2\varsigma +{\displaystyle \frac{1}{\varrho }}\left(\{\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]d_3+\mathsf{N}{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2\}{\varsigma }^2+\cdots \right),$$

(22)

and

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

(23)

Using a few fundamental mathematical steps, we rewrite Equation (20) as

$$1+{\displaystyle \frac{1}{\varrho }}\left(\tau \left({\displaystyle \frac{{\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{{\mathcal{G}}^{\prime }\left(w\right)}}\right)+(1-\tau )\left({\displaystyle \frac{w{\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }}{\mathcal{G}\left(w\right)}}\right)-1\right)=1-{\displaystyle \frac{\mathsf{L}}{\varrho }}{\Gamma }_2[{\eta }_1;{\kappa }_1]d_2\varsigma +$$

$${\displaystyle \frac{1}{\varrho }}\left(\{\mathsf{M}(2{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2-{\Gamma }_3[{\eta }_1;{\kappa }_1]d_3)+\mathsf{N}{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2\}{\varsigma }^2+\cdots \right),$$

(24)

and

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

(25)

where $\mathsf{L}$, $\mathsf{M}$, and $\mathsf{N}$ are as mentioned in (18).

By comparing the coefficients of terms with the same degree in Equations (22) and (23) and using Equation (19), we draw following conclusions.

$$\mathsf{L}{\Gamma }_2[{\eta }_1;{\kappa }_1]d_2=\varrho {\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{l}}_1,$$

(26)

$$\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]d_3+\mathsf{N}{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2=\varrho \left[{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{l}}_2+{\displaystyle \frac{{\mathcal{B}}_{q,2}\left(\varkappa \right)}{{\left[2\right]}_q}}{\mathfrak{l}}_1^2\right].$$

(27)

Likewise, using the equality (20), we compare the terms of the same degree in (24) and (25) to obtain

$$-\mathsf{L}{\Gamma }_2[{\eta }_1;{\kappa }_1]d_2=\varrho {\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{m}}_1$$

(28)

and

$$\mathsf{M}(2{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2-{\Gamma }_3[{\eta }_1;{\kappa }_1]d_3)+\mathsf{N}{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2=\varrho \left[{\mathcal{B}}_{q,1}\left(\varkappa \right){\mathfrak{m}}_2+{\displaystyle \frac{{\mathcal{B}}_{q,2}\left(\varkappa \right)}{{\left[2\right]}_q}}{\mathfrak{m}}_1^2\right].$$

(29)

From (26) and (28), we get

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

(30)

and

$$2{\mathsf{L}}^2{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2={\varrho }^2({\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2){\mathcal{B}}_{q,1}^2\left(\varkappa \right).$$

(31)

Addition of (27) and (29) yield

$$2(\mathsf{M}+\mathsf{N}){\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{d}_2^2=\varrho {\mathcal{B}}_{q,1}\left(\varkappa \right)({\mathfrak{l}}_2+{\mathfrak{m}}_2)+{\displaystyle \frac{\varrho {\mathcal{B}}_{q,2}\left(\varkappa \right)}{{\left[2\right]}_q}}({\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2).$$

(32)

Replacing ${\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2$ from (31) in (32), we get

$$d_2^2={\displaystyle \frac{{\varrho }^2{\mathcal{B}}_{q,1}^3\left(\varkappa \right)({\mathfrak{l}}_2+{\mathfrak{m}}_2)}{2\left[(\mathsf{M}+\mathsf{N}){\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2\varrho {\mathcal{B}}_{q,1}^2\left(\varkappa \right)-\frac{{\mathsf{L}}^2}{{\left[2\right]}_q}{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2{\mathcal{B}}_{q,2}\left(\varkappa \right)\right]}}.$$

(33)

Utilizing (9) for ${\mathcal{B}}_{q,1}\left(\varkappa \right)$, and (10) for ${\mathcal{B}}_{q,2}\left(\varkappa \right)$, and applying (21) to ${\mathfrak{l}}_2$, ${\mathfrak{m}}_2$ produces (14).

Subtracting equation (29) from (27), we obtain a bound on $|d_3|$:

$$d_3={\displaystyle \frac{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}{{\Gamma }_3[{\eta }_1;{\kappa }_1]}}{d}_2^2+{\displaystyle \frac{\varrho {\mathcal{B}}_{q,1}\left(\varkappa \right)({\mathfrak{l}}_2-{\mathfrak{m}}_2)}{2\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}}.$$

(34)

If we replace $d_2^2$ from (31) in (34) we get

$$d_3={\displaystyle \frac{{\varrho }^2{\mathcal{B}}_{q,1}^2\left(\varkappa \right)({\mathfrak{l}}_1^2+{\mathfrak{m}}_1^2)}{2{\Gamma }_3[{\eta }_1;{\kappa }_1]{\mathsf{L}}^2}}+{\displaystyle \frac{\varrho {\mathcal{B}}_{q,1}\left(\varkappa \right)({\mathfrak{l}}_2-{\mathfrak{m}}_2)}{2\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}}.$$

(35)

Equation (15) is derived from (35) by applying Equations (9) and (21).

Finally, we determine the bound for $|d_3-\xi d_2^2|$ using the values of $d_2^2$ and $d_3$ from (33) and (34), respectively. Consequently, we have

$$|d_3-\xi d_2^2|={\displaystyle \frac{\left|\varrho \right||{\mathcal{B}}_{q,1}\left(\varkappa \right)|}{2}}\left|\left({\displaystyle \frac{1}{\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+\mathcal{V}(\xi ,\varkappa )\right){\mathfrak{l}}_2-\left({\displaystyle \frac{1}{\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}}-\mathcal{V}(\xi ,\varkappa )\right){\mathfrak{m}}_2\right|,$$

where

$$\mathcal{V}(\xi ,\varkappa )={\displaystyle \frac{(\frac{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}{{\Gamma }_3[{\eta }_1;{\kappa }_1]}-\xi )\varrho {\mathcal{B}}_{q,1}^2\left(\varkappa \right)}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2\left[\varrho (\mathsf{M}+\mathsf{N}){\mathcal{B}}_{q,1}^2\left(\varkappa \right)-\frac{{\mathsf{L}}^2}{{\left[2\right]}_q}{\mathcal{B}}_{q,2}\left(\varkappa \right)\right]}}.$$

Clearly

$$|d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right||{\mathcal{B}}_{q,1}\left(\varkappa \right)|}{\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;\left|\mathcal{V}(\xi ,\varkappa )\right|\le {\displaystyle \frac{1}{\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill \\ \left|\varrho \right||{\mathcal{B}}_{q,1}\left(\varkappa \right)\left|\right|\mathcal{V}(\xi ,\varkappa )|\hfill & ;\left|\mathcal{V}(\xi ,\varkappa )\right|\ge {\displaystyle \frac{1}{\mathsf{M}{\Gamma }_3[{\eta }_1;{\kappa }_1]}}.\hfill \end{array}\right.$$

(36)

We derive (16) from (36), where ℸ is the same as in (17). □

Taking $p=2,s=1,{\eta }_1=a,{\eta }_2=b,$ and ${\kappa }_1=c$ in Theorem 1, we get the following result:

Corollary 1.

Let $\varkappa ,\xi \in \mathbb{R}$, $\nu \ge 1$, $0\le \delta \le 1$, and $0\le \tau \le 1$. If $\varphi \in \mathcal{A}$ is assigned to the family ${\mathfrak{T}}_{\sigma }^{2,1}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\Gamma }_2[a,b;c]}}\sqrt{{\displaystyle \frac{{|\left[2\right]}_q\varkappa -q|}{{\left[2\right]}_q|\varrho (\mathsf{M}+\mathsf{N}){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{\mathsf{L}}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{\Gamma }_3[a,b;c]{\mathsf{L}}^2}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q\mathsf{M}{\Gamma }_3[a,b;c]}},$$

and

$$\begin{array}{c}\hfill |d_3-\xi d_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q\mathsf{M}{\Gamma }_3[a,b;c]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[a,b;c]}{{\left({\Gamma }_2[a,b;c]\right)}^2}}\xi |\le \daleth \hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[a,b;c]}{{\left({\Gamma }_2[a,b;c]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[a,b;c]|\varrho (\mathsf{M}+\mathsf{N}){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{\mathsf{L}}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[a,b;c]}{{\left({\Gamma }_2[a,b;c]\right)}^2}}\xi |\ge \daleth ,\hfill \end{array}\right.\end{array}$$

where ℸ is as given by (17), ${\Gamma }_j[a,b;c]={\displaystyle \frac{{\left(a\right)}_{j-1}{\left(b\right)}_{j-1}}{{\left(c\right)}_{j-1}}}{\displaystyle \frac{1}{(j-1)!}}$ (see (6)), $\mathsf{L}$, $\mathsf{M}$, and $\mathsf{N}$ are as stated in (18).

Remark 3.

When $q\to 1^{-}$ and $\delta =1$, the two results obtained above reduce to the corresponding results established in the paper entitled ’Certain subfamilies of Te-univalent functions subordinate to Bernoulli polynomials’ by Swamy, Frasin, and Lupas, which has been submitted for publication.

3. Particular Cases

Using the above definition, we can obtain a number of subfamilies of ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,\delta ,\varkappa )$ associated with q-Bernoulli polynomials for specific parameters, such as $\delta$, $\tau$ and $\nu$. Consequently, the associated results are derived from the findings presented in the publication. We go over a few of these in this section.

Specializing Theorem 1 to the case $\tau =0$, we obtain the following result:

Corollary 2.

Let $\varkappa ,\xi \in \mathbb{R}$, $\nu \ge 1$, and $0\le \delta \le 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,\delta ,\varkappa )$, $\varkappa \in \mathbb{R}$, then

$$|d_2|\le$$

$${\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho ((2\nu +1)(\nu -\delta )+{\delta }^2){({\left[2\right]}_q\varkappa -q)}^2-\left[2_q\right]{(2\nu -\delta )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2\nu -\delta )}^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\nu -\delta ){\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

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

where

$${\daleth }_1=\left|{\displaystyle \frac{\varrho ((2\nu +1)(\nu -\delta )+{\delta }^2){({\left[2\right]}_q\varkappa -q)}^2-\left[2_q\right]{(2\nu -\delta )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{\varrho (3\nu -\delta ){({\left[2\right]}_q\varkappa -q)}^2}}\right|.$$

If we take $\tau =1$ in Theorem 1, we arrive at the following corollary.

Corollary 3.

Let $\varkappa ,\xi \in \mathbb{R}$, $\nu \ge 1$, and $0\le \delta \le 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,\delta ,\varkappa )$, $\varkappa \in \mathbb{R}$, then

$$|d_2|\le {\displaystyle \frac{1}{{\Gamma }_2[{\eta }_1;{\kappa }_1]}}$$

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

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{(2\nu -\delta )}^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\nu -\delta ){\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\nu -\delta ){\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\le {\daleth }_2\hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[{\eta }_1;{\kappa }_1]|\varrho ((8\nu +1)(\nu -\delta )+4{\delta }^2-2\delta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\nu -\delta )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\ge {\daleth }_2,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_2=\left|{\displaystyle \frac{\varrho ((8\nu +1)(\nu -\delta )+4{\delta }^2-2\delta ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\nu -\delta )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{3\varrho (3\nu -\delta ){({\left[2\right]}_q\varkappa -q)}^2}}\right|.$$

When $\nu$ is set to 1 in Theorem 1, the following corollary follows.

Corollary 4.

Let $\varkappa ,\xi \in \mathbb{R}$, $0\le \tau \le 1$, and $0\le \delta \le 1$. If $\varphi \in \mathcal{A}$ is a member of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,1,\delta ,\varkappa )$, then

$$|d_2|\le$$

$${\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho ((3-\delta ){\gamma }_3-\delta (2-\delta ){\gamma }_2){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2-\delta )}^2{\gamma }_1^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2-\delta )}^2{\gamma }_1^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3-\delta ){\gamma }_3{\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

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

where

$${\daleth }_3=\left|{\displaystyle \frac{\varrho ((3-\delta ){\gamma }_3-\delta (2-\delta ){\gamma }_2){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2-\delta )}^2{\gamma }_1^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{\varrho (3-\delta ){\gamma }_3{({\left[2\right]}_q\varkappa -q)}^2}}\right|,$$

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

(37)

Specializing Theorem 1 to $\delta =0$, we obtain the following result:

Corollary 5.

Let $\varkappa ,\xi \in \mathbb{R}$, $0\le \tau \le 1$, and $\nu \ge 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,0,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho \nu (1+2\nu +6\nu \tau ){({\left[2\right]}_q\varkappa -q)}^2-\left[2_q\right]4{\nu }^2{(1+\tau )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{\nu }^2{(1+\tau )}^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\nu (1+2\tau ){\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

$$\begin{array}{c}\hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_{q}3\nu (1+2\tau ){\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\le {\daleth }_4\hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[{\eta }_1;{\kappa }_1]|\varrho \nu (1+2\nu +6\nu \tau ){({\left[2\right]}_q\varkappa -q)}^2-\left[2_q\right]4{\nu }^2{(1+\tau )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\ge {\daleth }_4,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_4=\left|{\displaystyle \frac{\varrho (1+2\nu +6\nu \tau ){({\left[2\right]}_q\varkappa -q)}^2-\left[2_q\right]4\nu {(1+\tau )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{3\varrho (1+2\tau ){({\left[2\right]}_q\varkappa -q)}^2}}\right|.$$

The important cases from the family ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,0,\varkappa )$ corresponding to $\tau =0$ and $\tau =1$ are obtained as follows.

Case 1. Let $\tau =0$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,0,\varkappa )$, and $\nu \ge 1,$ is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }{)}^{\nu }-1\right)\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),and1+{\displaystyle \frac{1}{\varrho }}\left({\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),$$

where $\varsigma ,w\in \mathfrak{U}$.

Case 2. Let $\tau =1$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,0,\varkappa )$, $\varkappa \in \mathbb{R},\varrho \in \mathbb{C}\setminus \left\{0\right\}$, and $\nu \ge 1,$ is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left({\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }-1\right)\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),and1+{\displaystyle \frac{1}{\varrho }}\left({\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),$$

where $\varsigma ,w\in \mathfrak{U}$.

The initial coefficient estimates and FSF, for functions belonging to the families ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,0,\varkappa )$ and ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,0,\varkappa )$ are presented in the following two corollaries.

Corollary 6.

Let $\varkappa \in \mathbb{R}$, and $\nu \ge 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,0,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho \nu (1+2\nu ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{\nu }^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{\nu }^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q\nu {\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

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

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

$$\begin{array}{c}\hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_{q}3\nu {\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\le {\daleth }_5\hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[{\eta }_1;{\kappa }_1]|\varrho \nu (1+2\nu ){({\left[2\right]}_{q}x-q)}^2-4{\left[2\right]}_q{\nu }^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\ge {\daleth }_5,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_5=\left|{\displaystyle \frac{\varrho (1+2\nu ){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q\nu ({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{3\varrho {({\left[2\right]}_q\varkappa -q)}^2}}\right|.$$

Corollary 7.

Let $\varkappa ,\xi \in \mathbb{R}$, and $\nu \ge 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,0,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho \nu (1+8\nu ){({\left[2\right]}_q\varkappa -q)}^2-16{\left[2\right]}_q{\nu }^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{16{\left[2\right]}_q^2{\nu }^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{9{\left[2\right]}_q\nu {\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

$$\begin{array}{c}\hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{9{\left[2\right]}_{q}3\nu {\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\le {\daleth }_6\hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[{\eta }_1;{\kappa }_1]|\varrho \nu (1+8\nu ){({\left[2\right]}_q\varkappa -q)}^2-16{\left[2\right]}_q{\nu }^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\ge {\daleth }_6,\hfill \end{array}\right.\end{array}$$

where

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

Specializing Theorem 1 to $\delta =1$, we obtain the following result:

Corollary 8.

Let $\varkappa ,\xi \in \mathbb{R}$, $0\le \tau \le 1$, and $\nu \ge 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,1,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho V{({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\nu -1)}^2{(1+\tau )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2\nu -1)}^2{(1+\tau )}^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\nu -1)(1+2\tau ){\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

$$\begin{array}{c}\hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\nu -1)(1+2\tau ){\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\le {\daleth }_7\hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[{\eta }_1;{\kappa }_1]|\varrho V{({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\nu -1)}^2{(1+\tau )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\ge {\daleth }_7,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_7=\left|{\displaystyle \frac{\varrho V{({\left[2\right]}_q\varkappa -q)}^2-\left[2_q\right]{(2\nu -1)}^2{(1+\tau )}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{\varrho (3\nu -1)(1+2\tau ){({\left[2\right]}_q\varkappa -q)}^2}}\right|,$$

and V = $\nu (\nu -1)(6\tau +1)+\tau +{\nu }^2$.

The important cases from the family ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,\tau ,\nu ,1,\varkappa )$ corresponding to $\tau =0$ and $\tau =1$ are obtained as follows.

Case 3. Let $\tau =0$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,1,\varkappa )$, $\varkappa \in \mathbb{R}$, and $\nu \ge 1,$ is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{\varsigma {\left({\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\nu }}{{\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)}}-1\right)\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),and1+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{w{\left({\mathcal{G}}^{\prime }\left(w\right)\right)}^{\nu }}{\mathcal{G}\left(w\right)}}-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),$$

where $\varsigma ,w\in \mathfrak{U}$.

Case 4. Let $\tau =1$. Then ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,1,\varkappa )$, $\varkappa \in \mathbb{R}$, and $\nu \ge 1,$ is a family of functions $\varphi \in \mathcal{A}$ satisfying

$$1+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{{\left[{\left(\varsigma {\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }\right)}^{\prime }\right]}^{\nu }}{{\left({\mathcal{H}}_{p,s}[{\eta }_1;{\kappa }_1]\varphi \left(\varsigma \right)\right)}^{\prime }}}-1\right)\prec {\mathfrak{B}}_q(\varkappa ,\varsigma ),\varsigma \in \mathfrak{U},$$

and

$$1+{\displaystyle \frac{1}{\varrho }}\left({\displaystyle \frac{{\left[{\left(w{\mathcal{G}}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\nu }}{{\mathcal{G}}^{\prime }\left(w\right)}}-1\right)\prec {\mathfrak{B}}_q(\varkappa ,w),w\in \mathfrak{U}.$$

The initial coefficient estimates and the functional $|d_3-\xi d_2^2|,\xi \in \mathbb{R}$, for functions belonging to the families ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,1,\varkappa )$ and ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,1,\varkappa )$ are established in the following two corollaries.

Corollary 9.

Let $\varkappa ,\xi \in \mathbb{R}$, and $\nu \ge 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,0,\nu ,1,\varkappa )$, then

$$|d_2|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho \nu (2\nu -1){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\nu -1)}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{{\left[2\right]}_q^2{(2\nu -1)}^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\nu -1){\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{{\left[2\right]}_q(3\nu -1){\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\le {\daleth }_8\hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[{\eta }_1;{\kappa }_1]|\varrho \nu (2\nu -1){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\nu -1)}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\ge {\daleth }_8,\hfill \end{array}\right.\end{array}$$

where

$${\daleth }_8=\left|{\displaystyle \frac{\varrho \nu (2\nu -1){({\left[2\right]}_q\varkappa -q)}^2-{\left[2\right]}_q{(2\nu -1)}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})}{\varrho (3\nu -1){({\left[2\right]}_q\varkappa -q)}^2}}\right|.$$

Corollary 10.

Let $\varkappa ,\xi \in \mathbb{R}$, and $\nu \ge 1$. If $\varphi \in \mathcal{A}$ is an element of the class ${\mathfrak{T}}_{\sigma }^{p,s}(\varrho ,1,\nu ,1,\varkappa )$, then

$$|d_2|\le$$

$${\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q)|\sqrt{{|\left[2\right]}_q\varkappa -q|}}{{\Gamma }_2[{\eta }_1;{\kappa }_1]\sqrt{{\left[2\right]}_q|\varrho (8{\nu }^2-7\nu +1){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\nu -1)}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}},$$

$$|d_3|\le {\displaystyle \frac{|\varrho ({\left[2\right]}_q\varkappa -q){|}^2}{4{\left[2\right]}_q^2{(2\nu -1)}^2{\Gamma }_3[{\eta }_1;{\kappa }_1]}}+{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\nu -1){\Gamma }_3[{\eta }_1;{\kappa }_1]}},$$

and

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

$$\begin{array}{c}\hfill \le \left\{\begin{array}{cc}{\displaystyle \frac{\left|\varrho \right({\left[2\right]}_q\varkappa -q\left)\right|}{3{\left[2\right]}_q(3\nu -1){\Gamma }_3[{\eta }_1;{\kappa }_1]}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\le {\daleth }_9\hfill \\ {\displaystyle \frac{{\left|\varrho \right|}^2{|\left[2\right]}_q{\varkappa -q|}^3|1-\frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}\xi |}{{\left[2\right]}_q{\Gamma }_3[{\eta }_1;{\kappa }_1]|\varrho (8{\nu }^2-7\nu +1){({\left[2\right]}_q\varkappa -q)}^2-4{\left[2\right]}_q{(2\nu -1)}^2({\varkappa }^2-\varkappa +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q})|}}\hfill & ;|1-{\displaystyle \frac{{\Gamma }_3[{\eta }_1;{\kappa }_1]}{{\left({\Gamma }_2[{\eta }_1;{\kappa }_1]\right)}^2}}\xi |\ge {\daleth }_9,\hfill \end{array}\right.\end{array}$$

where

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

Remark 4.

When $q\to 1^{-}$ and $\delta =1$, Corollaries 9, 10, and 4 reduce to Corollaries 3.1, 3.2, and 3.3, respectively, in the paper entitled “Certain subfamilies of Te-univalent functions subordinate to Bernoulli polynomials” by Swamy, Frasin, and Lupas, which has been submitted for publication.

4. Conclusions

In this paper, we derived explicit upper bounds for the initial coefficients $|d_2|$ and $|d_3|$, of analytic functions belonging to newly introduced subclasses associated with q-Bernoulli polynomials. In addition, we established general bounds for the FSF $|d_3-\xi d_2^2|,\xi \in \mathbb{R}$, thereby encompassing a broad spectrum of function classes within this novel framework.

By specializing the involved parameters, as detailed in Section 3, our results reduce to several new and previously unreported coefficient estimates and functional inequalities. These special cases highlight the adaptability and strength of our theoretical findings while contributing fresh perspectives to the literature on Te-univalent functions. Connections to previously known results have been underscored to demonstrate the consistency and relevance of our work.

Nonetheless, the subclasses considered here represent only a portion of the vast landscape of ${\mathcal{T}}_{\sigma }^{p,s}[{\eta }_1;{\kappa }_1]$ families available in the literature. Future work can extend this analysis to other important subclasses and explore the application of additional orthogonal polynomial families, such as Euler and Hermite polynomials, thereby enriching the theory further and broadening the scope of Geometric Function Theory.