Novel Bi-Univalent Subclasses Generated by the q-Analogue of the Ruscheweyh Operator and Hermite Polynomials

Abstract

This work introduces new bi-univalent function classes defined using the fractional q-Ruscheweyh operator and characterized by subordination to q-Hermite polynomials. We derive coefficient bounds and Fekete–Szegö inequalities for these classes and show that our results generalize several earlier findings in both the classical and q-analytic settings. The approach highlights the effectiveness of q-Hermite structures in analyzing operator-defined subclasses of bi-univalent functions.

1. Introduction

The study of analytic and univalent functions has long been central to geometric function theory, yielding deep connections with complex analysis, operator theory, and geometric mappings in the unit disk. In recent decades, the study of q-calculus and its associated operators has attracted considerable attention in geometric function theory. In particular, the q-analogue of the Ruscheweyh operator, introduced by Aldweby and Darus [1], has been widely used to explore various properties of analytic and univalent functions. Researchers have further extended these ideas using fractional and convolution-based approaches [2,3,4], leading to new results in differential subordination and operator theory. This work builds on these developments to investigate new subclasses of analytic functions defined via the q-Ruscheweyh operator.

Let $\mathfrak{A}$ be the class of analytic functions in the unit disk $\mathbb{D}$ of the form

$$f\left(\zeta \right)=\zeta +\sum _{n=2}^{\infty }{a}_n{\zeta }^n.$$

(1)

Within $\mathfrak{A}$, the subclass $\mathcal{S}$ consists of normalized univalent functions satisfying $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$. Each $f\in \mathcal{S}$ has an inverse $f^{-1}$ defined in $\left|w\right|<r_0\left(f\right)(r_0\left(f\right)\ge 1/4)$, with the expansion

$$g\left(\mathsf{w}\right)=f^{-1}\left(\mathsf{w}\right)=\mathsf{w}-a_2{\mathsf{w}}^2+(2a_2^2-a_3){\mathsf{w}}^3-(5a_2^3-5a_2{a}_3+a_4){\mathsf{w}}^4+\dots .$$

(2)

Bi-univalent functions are of particular interest in geometric function theory because both a function f and its inverse $f^{-1}$ are univalent in the unit disk $\mathbb{D}$. We denote by ℵ the class of all such functions.

Some examples of bi-univalent functions and their inverses are as follows:

• $f_1\left(\zeta \right)={\displaystyle \frac{\zeta }{1-\zeta }}$ with inverse $f_1^{-1}\left(\mathsf{w}\right)={\displaystyle \frac{\mathsf{w}}{1+\mathsf{w}}}$.
• $f_2\left(\zeta \right)={\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\zeta }{1-\zeta }}\right)$ with inverse $f_2^{-1}\left(\mathsf{w}\right)={\displaystyle \frac{e^{2\mathsf{w}}-1}{e^{2\mathsf{w}}+1}}$.
• $f_3\left(\zeta \right)=-log(1-\zeta )$ with inverse $f_3^{-1}\left(\mathsf{w}\right)={\displaystyle \frac{e^{\mathsf{w}}-1}{e^{\mathsf{w}}}}$.

Another central concept is subordination, which provides a way to compare analytic functions. We say f is subordinate to g, written $f\left(\zeta \right)\prec g\left(\zeta \right),$ if there exists an analytic mapping $w:\mathbb{D}\to \mathbb{D}$ with $w\left(0\right)=0$ and $\left|w\right(\zeta \left)\right|<1$ such that $f\left(\zeta \right)=g\left(w\right(\zeta \left)\right)$. If g is univalent, subordination reduces to the intuitive condition $f\left(0\right)=g\left(0\right)$ and $f\left(\mathbb{D}\right)\subset g\left(\mathbb{D}\right)$, providing a clear geometric interpretation of inclusion [5,6,7].

Lewin [8] initiated the study of the bi-univalent class ℵ, showing that $|a_2|<1.51$. Later works by Brannan and Clunie [9,10,11] and Netanyahu [12] extended these investigations, providing bounds for $|a_2|$ and $|a_3|$ in various subclasses. Despite these advances, estimating $|a_n|$ for $n\ge 4$ remains an open problem. Over the decades, many subclasses have been analyzed, and sharp or non-sharp bounds for the first few coefficients have been established in multiple contexts [13,14,15,16].

Ma and Minda [17] introduced a flexible framework using an analytic function $\Psi$ with

$$\Re (\Psi \left(\zeta \right))>0,\Psi \left(0\right)=1,{\Psi }^{\prime }\left(0\right)>0,$$

mapping the unit disk $\mathbb{D}$ into a starlike domain with respect to 1 and symmetric about the real axis. This yields the generalized classes:

$${\mathcal{S}}^{*}(\Psi )=\left\{f\in \mathfrak{A}:{\displaystyle \frac{z{f}^{\prime }\left(\zeta \right)}{f\left(\zeta \right)}}\prec \Psi \left(\zeta \right)\right\},\mathcal{C}(\Psi )=\left\{f\in \mathfrak{A}:1+{\displaystyle \frac{z{f}^{\prime \prime }\left(\zeta \right)}{f^{\prime }\left(\zeta \right)}}\prec \Psi \left(\zeta \right)\right\}.$$

Several notable choices of $\Psi$ and the corresponding subclasses are summarized in

Table 1. Representative subclasses generated via Ma–Minda functions $\Psi$.

$\Psi \left(\mathit{\zeta }\right)$	Subclasses (References)
$\sqrt{1+\zeta }$	${\mathcal{S}}_L^{*}$ (Sokoł, 2009 [18]; Raza and Malik, 2013 [19])
$\zeta +\sqrt{1+{\zeta }^2}$	${\mathcal{S}}_l^{*}$ (Raina and Sokoł, 2015 [20])
$1+\zeta -\frac{1}{3}{\zeta }^2+\frac{1}{9}{\zeta }^3$	${\mathcal{S}}_H^{*}$ (Tayyah et al., 2025 [21])
$1+sin\left(\zeta \right)$	${\mathcal{S}}_{sin}^{*}$ (Arif et al., 2019 [22])

, highlighting the diversity and flexibility of the Ma–Minda framework.

In geometric function theory, starlike and convex functions are often characterized using subordination. A function f is starlike of order $\epsilon$ if

$${\displaystyle \frac{z{f}^{\prime }\left(\zeta \right)}{f\left(\zeta \right)}}\prec {\displaystyle \frac{1+(1-2\epsilon )\zeta }{1-\zeta }},0\le \epsilon <1,$$

and convex of order $\epsilon$ if

$$1+{\displaystyle \frac{z{f}^{\prime \prime }\left(\zeta \right)}{f^{\prime }\left(\zeta \right)}}\prec {\displaystyle \frac{1+(1-2\epsilon )\zeta }{1-\zeta }}.$$

We begin by presenting the essential definitions and key concepts associated with the applications of q-calculus. Definitions are first given for fractional q-calculus operators in a complex-valued function $f\left(\zeta \right)$, as follows:

Definition 1 

([23]). Let $0<q<1$. The q-number ${\left[n\right]}_q$ is defined by

$${\left[n\right]}_q=\left\{\begin{array}{cc}{\displaystyle \frac{1-q^n}{1-q}},\hfill & n\in \mathbb{C},\hfill \\ 1+q+q^2+\dots +q^{m-1},\hfill & n\in \mathbb{N}.\hfill \end{array}\right.$$

(3)

Definition 2. 

Let $0<q<1$. The q-factorial ${\left[n\right]}_q!$ is defined by

$${\left[n\right]}_q!=\left\{\begin{array}{cc}{\left[n\right]}_q{[n-1]}_q\dots {\left[1\right]}_q,\hfill & n=1,2,\dots ,\hfill \\ 1,\hfill & n=0.\hfill \end{array}\right.$$

(4)

Definition 3 

([24]). Let $f\in \mathfrak{A}$ and $0<q<1$. The q-derivative operator of a function f is defined by

$${\partial }_{q}f\left(\zeta \right)=\left\{\begin{array}{cc}{\displaystyle \frac{f\left(q\zeta \right)-f\left(\zeta \right)}{(q-1)\zeta }},\hfill & z\ne 0,\hfill \\ f^{\prime }\left(\zeta \right),\hfill & z=0.\hfill \end{array}\right.$$

(5)

We note from Definition (3) that

$$\underset{q\to 1}{lim}\left({\partial }_{q}f\right)\left(\zeta \right)=\underset{q\to 1}{lim}{\displaystyle \frac{f\left(q\zeta \right)-f\left(\zeta \right)}{(q-1)\zeta }}=f^{\prime }\left(\zeta \right).$$

From Equations (3) and (5), we get

$${\partial }_{q}f\left(\zeta \right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{\zeta }^{n-1}.$$

In 2014, Aldweby and Darus [1] introduced and studied the q-analogue of the Ruscheweyh operator ${\mathcal{R}}_{\gamma }^q$ by

$${\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)=\zeta +\sum _{n=2}^{\infty }{\displaystyle \frac{{[n+\gamma -1]}_q!}{{\left[\gamma \right]}_q!{[n-1]}_q!}}{a}_n{\zeta }^n=\zeta +\sum _{n=2}^{\infty }{\mathbb{J}}_{\gamma }^q\left(n\right)a_n{\zeta }^n,$$

(6)

where ${\mathbb{J}}_{\gamma }^q\left(n\right)={\displaystyle \frac{{[n+\gamma -1]}_q!}{{\left[\gamma \right]}_q!{[n-1]}_q!}},\gamma >-1$ and ${\left[n\right]}_q!$ is given by Equation (4).

Remark 1. 

The operator ${\mathcal{R}}_{\gamma }^q$ defined in (6) is referred to as a fractional q–Ruscheweyh operator since the order parameter γ is allowed to take non-integer values. When $\gamma \in \mathbb{N}$, the operator reduces to the classical integer-order q–Ruscheweyh derivative, whereas for $\gamma \notin \mathbb{N}$ it yields a genuine fractional extension in the sense of convolution-based operators commonly used in geometric function theory.

Moreover, as $q\to 1$, we have

$$\underset{q\to 1}{lim}{\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)={\mathcal{R}}_{\gamma }f\left(\zeta \right)=\zeta +\sum _{n=2}^{\infty }{\displaystyle \frac{(n+\gamma -1)!}{\gamma !(n-1)!}}{a}_n{\zeta }^n,$$

where ${\mathcal{R}}_{\gamma }f\left(\zeta \right)$ is the classical Ruscheweyh differential operator introduced in [25], and it has been studied by several authors (see [26,27,28]).

In particular, when $\gamma =0$, the operator reduces to the identity operator

$${\mathcal{R}}_{0}f\left(\zeta \right)=\zeta +\sum _{n=2}^{\infty }{a}_n{\zeta }^n=f\left(\zeta \right),$$

which coincides with the function f itself.

The q-derivative is linear and satisfies q-versions of the product and quotient rules. Higher-order derivatives follow the q-Leibniz formula:

$${\partial }_q^{\left(n\right)}\left(fg\right)\left(\zeta \right)=\sum _{k=0}^n{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)}_q{\partial }_q^{\left(k\right)}f\left(q^{n-k}\zeta \right){\partial }_q^{(n-k)}g\left(q^k\zeta \right).$$

Using ${\partial }_q$, q-starlike and q-convex functions are defined as

$$S_q^{*}=\left\{f\in \mathfrak{A}:\Re \left({\displaystyle \frac{\zeta {\partial }_{q}f\left(\zeta \right)}{f\left(\zeta \right)}}\right)>0\right\},K_q=\left\{f\in \mathfrak{A}:\Re \left({\displaystyle \frac{{\partial }_q\left(\zeta {\partial }_{q}f\left(\zeta \right)\right)}{{\partial }_{q}f\left(\zeta \right)}}\right)>0\right\},$$

(7)

which reduce to the classical starlike and convex classes as $q\to 1^{-}$.

In terms of subordination, we have

$$S_q^{*}(\Psi )=\left\{f\in \mathfrak{A}:{\displaystyle \frac{\zeta {\partial }_{q}f\left(\zeta \right)}{f\left(\zeta \right)}}\prec \Psi \left(\zeta \right)\right\},K_q(\Psi )=\left\{f\in \mathfrak{A}:{\displaystyle \frac{{\partial }_q\left(\zeta {\partial }_{q}f\left(\zeta \right)\right)}{{\partial }_{q}f\left(\zeta \right)}}\prec \Psi \left(\zeta \right)\right\},$$

(8)

providing a generalized framework for q-analogs of starlike and convex functions [29,30].

The q-Hermite polynomials, originally introduced by Rogers [31] (see also [32,33,34,35]), play a central role in the theory of q-orthogonal polynomials and special functions. They are defined via the generating function:

$${\mathcal{K}}_k\left(s\right|q)=\sum _{k=0}^{\infty }{H}_k(x;q)t^k{(q;q)}_k^{-1}=\prod _{k=0}^{\infty }{\displaystyle \frac{1}{1-2xt{q}^k+t^2{q}^{2k}}},0<q<1.$$

(9)

The q-derivative of the q-Hermite polynomial satisfies the relation

$${\partial }_q\left\{{\mathcal{K}}_{k+1}\left(s\right|q)\right\}={\left[k\right]}_q{\mathcal{K}}_k\left(s\right|q).$$

(10)

Furthermore, Ismail et al. [31] established the recurrence relation

$$t{\mathcal{K}}_k\left(s\right|q)={\mathcal{K}}_{k+1}\left(s\right|q)+{\left[k\right]}_q{\mathcal{K}}_{k-1}\left(s\right|q),$$

(11)

with the initial conditions

$${\mathcal{K}}_0\left(s\right|q)=1,{\mathcal{K}}_{-1}\left(s\right|q)=0.$$

From (9), the first few polynomials are obtained as

$$\begin{array}{cc}\hfill {\mathcal{K}}_1\left(s\right|q)& =s,\hfill \\ \hfill {\mathcal{K}}_2\left(s\right|q)& =s^2-1,\hfill \\ \hfill {\mathcal{K}}_3\left(s\right|q)& =s^3-(2+q)s,\hfill \\ \hfill {\mathcal{K}}_4\left(s\right|q)& =s^4-(3+2q+q^2)s^2+(1+q+q^2).\hfill \end{array}$$

These relations highlight the elegant recursive structure of the q-Hermite family, smoothly converging to the classical Hermite polynomials as $q\to 1^{-}$.

Next, we introduce the q-Babalola convolution operator, which will be central in the following definitions. Let

$$\mathcal{M}(\zeta ,s,q)=\sum _{k=0}^{\infty }{\mathcal{K}}_k\left(s\right|q){\zeta }^k.$$

Remark 2. 

The q-Hermite polynomials unify several classical families: for $q=1$, we recover the standard Hermite polynomials, ${\mathcal{K}}_k\left(s\right|1)={\mathcal{K}}_k^{\mathsf{v}}\left(s\right)$, while for $q=0$, they reduce to the Chebyshev polynomials of the first kind, $U_k(s/2)$, which satisfy the recursion

$$2s{U}_k\left(s\right)=U_{k-1}\left(s\right)+U_{k+1}\left(s\right),U_0\left(s\right)=1,U_{-1}\left(s\right)=0.$$

Remark 3. 

The generating function $\mathcal{M}(\zeta ,s,q)$ associated with q-Hermite polynomials maps the unit disk onto domains symmetric with respect to the real axis. For $s\in (0.5, 1)$, the function has a positive real part, ensuring starlikeness of the image domain. As q varies, the geometry of the image domain deforms continuously while preserving symmetry.

Lemma 1. 

Let $p\left(\zeta \right)=1+{\sum }_{k=1}^{\infty }{\mathsf{b}}_k{\zeta }^k$ be an analytic function in the unit disk $\mathbb{U}$. Then the coefficients satisfy

$$|{\mathsf{b}}_k|\le 2,k\ge 1.$$

(12)

2. Bounds for Initial Coefficients and Fekete–Szegö Inequalities in the Class ${\mathbb{S}}_{\aleph }(\mathit{s},\mathit{q},\mathit{\gamma })$

In this section, we introduce the class of starlike bi-univalent functions defined via the q-Ruscheweyh operator and subordinated to q-Hermite polynomials. We then establish sharp bounds for the initial coefficients of functions in this class and derive the corresponding Fekete–Szegö inequalities. These results provide insight into the geometric properties and coefficient behavior of the newly defined subclasses.

Definition 4. 

A function $f\in \aleph$ given by (1) belongs to the starlike class ${\mathbb{S}}_{\aleph }(s,q,\gamma )$ if

$${\displaystyle \frac{\zeta {\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)}}\prec \mathcal{M}(\zeta ,s,q),$$

(13)

and

$${\displaystyle \frac{\mathsf{w}{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)}}\prec \mathcal{M}(\mathsf{w},s,q),$$

(14)

where g is the inverse function of f.

Example 1. 

Let $\omega \left(\zeta \right)=\zeta$ and define

$$f\left(\zeta \right)=\zeta +{\displaystyle \frac{\mathcal{M}(1,s,q))}{{[\gamma +1]}_q}}{\zeta }^2+\dots .$$

Then f satisfies the subordination condition (13) and hence belongs to the class ${\mathbb{S}}_{\aleph }(s,q,\gamma )$. Its inverse function is given by

$$f^{-1}\left(\mathsf{w}\right)=\mathsf{w}-{\displaystyle \frac{\mathcal{M}(1,s,q)}{{[\gamma +1]}_q}}{\mathsf{w}}^2+\dots ,$$

which satisfies the inverse subordination condition (14).

Example 2. 

If $q\to 1$, then the class ${\mathbb{S}}_{\aleph }(s,1,\gamma )$ reduces to the classical starlike class ${\mathbb{S}}_{\aleph }(s,\gamma )$ defined by

$${\displaystyle \frac{\zeta {\left({\mathcal{R}}_{\gamma }f\left(\zeta \right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }f\left(\zeta \right)}}\prec \mathcal{M}(\zeta ,s,1),and{\displaystyle \frac{\mathsf{w}{\left({\mathcal{R}}_{\gamma }g\left(\mathsf{w}\right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }g\left(\mathsf{w}\right)}}\prec \mathcal{M}(\mathsf{w},s,1),$$

where ${\mathcal{R}}_{\gamma }f\left(\zeta \right)$ is the classical Ruscheweyh differential operator.

Example 3. 

If $\gamma =0$ and $q\to 1$, then the class ${\mathbb{S}}_{\aleph }(s,1,0)$ reduces to the class ${\mathbb{S}}_{\aleph }\left(s\right)$, that is,

$${\displaystyle \frac{z{f}^{\prime }\left(\zeta \right)}{f\left(\zeta \right)}}\prec \mathcal{M}(\zeta ,s,q),and{\displaystyle \frac{\mathsf{w}{g}^{\prime }\left(\mathsf{w}\right)}{g\left(\mathsf{w}\right)}}\prec \mathcal{M}(\mathsf{w},s,q),$$

where ${\mathcal{R}}_0^{1}f\left(\zeta \right)=f\left(\zeta \right)$ and ${\mathcal{R}}_0^{1}g\left(\mathsf{w}\right)=g\left(\mathsf{w}\right)$.

Remark 4. 

The restriction $\gamma \notin \mathbb{N}$ is imposed to avoid reduction to the classical integer-order q–Ruscheweyh operator, for which similar coefficient estimates are already available in the literature. For $\gamma \in \mathbb{N}$, all results remain valid; however, they reduce to known special cases and therefore do not yield new contributions.

Theorem 1. 

Assume that $0<q<1$; the order γ is non-integer. For a function $f\in \aleph$ of the form (1) belonging to ${\mathbb{S}}_{\aleph }(s,q,\gamma )$, the following coefficient bounds hold:

$$|a_2|\le \sqrt{{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{\left|2{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2-\left({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)\right){\left(\frac{{\mathbb{J}}_{\gamma }^q\left(2\right)}{{\mathcal{K}}_1(s\mid q)}\right)}^2\right|}}},$$

(15)

and

$$|a_3|\le {\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2{\mathbb{J}}_{\gamma }^q\left(3\right)}}+{\left({\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{{\mathbb{J}}_{\gamma }^q\left(2\right)}}\right)}^2,$$

(16)

where

$${\mathbb{J}}_{\gamma }^q\left(2\right)={\displaystyle \frac{{[1+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[1\right]}_q!}},{\mathbb{J}}_{\gamma }^q\left(3\right)={\displaystyle \frac{{[2+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[2\right]}_q!}}.$$

Proof. 

Let $f\in \aleph$ be defined by (1) and assume that f belongs to the class ${\mathbb{S}}_{\aleph }(s,q,\gamma )$. Consider the functional relations

$${\displaystyle \frac{\zeta {\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)}}=\mathcal{M}(\beth \left(\zeta \right),s,q),{\displaystyle \frac{\mathsf{w}{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)}}=\mathcal{M}(\daleth \left(\mathsf{w}\right),s,q),$$

(17)

where g denotes the inverse of f. Let $p\left(\zeta \right),c\left(\mathsf{w}\right)\in \mathcal{P}$ be analytic functions defined as

$$p\left(\zeta \right)={\displaystyle \frac{1+\beth \left(\zeta \right)}{1-\beth \left(\zeta \right)}}=1+{\mathsf{p}}_1\zeta +{\mathsf{p}}_2{\zeta }^2+{\mathsf{p}}_3{\zeta }^3+\dots ,\beth \left(\zeta \right)={\displaystyle \frac{p\left(\zeta \right)-1}{p\left(\zeta \right)+1}},(\zeta \in \mathbb{D}),$$

(18)

and

$$c\left(\mathsf{w}\right)={\displaystyle \frac{1+\daleth \left(\mathsf{w}\right)}{1-\daleth \left(\mathsf{w}\right)}}=1+{\mathsf{v}}_1\mathsf{w}+{\mathsf{v}}_2{\mathsf{w}}^2+{\mathsf{v}}_3{\mathsf{w}}^3+\dots ,\daleth \left(\mathsf{w}\right)={\displaystyle \frac{c\left(\mathsf{w}\right)-1}{c\left(\mathsf{w}\right)+1}},(\mathsf{w}\in \mathbb{D}).$$

(19)

Expanding these functions in power series form gives the following:

$$\beth \left(\zeta \right)=\frac{1}{2}\left[{\mathsf{p}}_1\zeta +\left({\mathsf{p}}_2-\frac{{\mathsf{p}}_1^2}{2}\right){\zeta }^2+\left({\mathsf{p}}_3-{\mathsf{p}}_1{\mathsf{p}}_2+\frac{{\mathsf{p}}_1^3}{4}\right){\zeta }^3+\dots \right],$$

(20)

and

$$\daleth \left(\mathsf{w}\right)=\frac{1}{2}\left[{\mathsf{v}}_1\mathsf{w}+\left({\mathsf{v}}_2-\frac{{\mathsf{v}}_1^2}{2}\right){\mathsf{w}}^2+\left({\mathsf{v}}_3-{\mathsf{v}}_1{\mathsf{v}}_2+\frac{{\mathsf{v}}_1^3}{4}\right){\mathsf{w}}^3+\dots \right].$$

(21)

Using the generating function $\mathcal{M}(\zeta ,s,q)$ given by (9), we obtain the following:

$$\begin{array}{cc}\hfill \mathcal{M}(\beth (\zeta ),s,q)& =1+{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{p}}_{1}z+\left[{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{p}}_2-{\displaystyle \frac{{\mathsf{p}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{p}}_1^2\right]{\zeta }^2+\dots ,\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \mathcal{M}(\daleth (\mathsf{w}),s,q)& =1+{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{v}}_{1}w+\left[{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{v}}_2-{\displaystyle \frac{{\mathsf{v}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{v}}_1^2\right]{\mathsf{w}}^2+\dots .\hfill \end{array}$$

(23)

On the other hand, applying the q-operator ${\mathcal{R}}_{\gamma }^q$ to f and g yields the following series expansions:

$${\displaystyle \frac{\zeta {\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)}}=1+{\mathbb{J}}_{\gamma }^q\left(2\right)a_2\zeta +\left(-a_2^2{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2+2{\mathbb{J}}_{\gamma }^q\left(3\right)a_3\right){\zeta }^2+\dots ,$$

(24)

and

$${\displaystyle \frac{\mathsf{w}{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime }}{{\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)}}=1-{\mathbb{J}}_{\gamma }^q\left(2\right)a_{2}w+\left(a_2^2[4{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2]-2{\mathbb{J}}_{\gamma }^q\left(3\right)a_3\right){\mathsf{w}}^2+\dots .$$

(25)

A comparison of the coefficients of $\zeta$ in Equations (20) and (22), together with the coefficients of $\mathsf{w}$ in Equations (21) and (23), yields the following:

$${\mathbb{J}}_{\gamma }^q\left(2\right)a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{p}}_1,$$

(26)

$$2{\mathbb{J}}_{\gamma }^q\left(3\right)a_2-a_2^2{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{p}}_2-{\displaystyle \frac{{\mathsf{p}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{p}}_1^2,$$

(27)

$$-{\mathbb{J}}_{\gamma }^q\left(2\right)a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{v}}_1,$$

(28)

and

$$\left[4{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\right]a_2^2-2{\mathbb{J}}_{\gamma }^q\left(3\right)a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{v}}_2-{\displaystyle \frac{{\mathsf{v}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{v}}_1^2.$$

(29)

By referring to Equations (26) and (28), we obtain the following:

$${\mathsf{p}}_1=-{\mathsf{v}}_1$$

(30)

and

$$2{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2{a}_2^2={\left({\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\right)}^2\left({\mathsf{p}}_1^2+{\mathsf{v}}_1^2\right).$$

(31)

By adding Equations (27) and (29), we obtain the following:

$$2\left(2{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2\right)a_2^2=\left({\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}-{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{4}}\right)({\mathsf{p}}_1^2+{\mathsf{v}}_1^2)+{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}({\mathsf{p}}_2+{\mathsf{v}}_2).$$

(32)

Using Equation (31), we obtain the following:

$$a_2^2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)({\mathsf{p}}_2+{\mathsf{v}}_2)}{4\left[2{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2-({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)){\left(\frac{{\mathbb{J}}_{\gamma }^q\left(2\right)}{{\mathcal{K}}_1(s\mid q)}\right)}^2\right]}}.$$

(33)

Now, we determine an upper bound for $|a_2|$. By applying Lemma 1, we obtain the following:

$$|a_2|\le \sqrt{{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{|2{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2-\left({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)\right){\left(\frac{{\mathbb{J}}_{\gamma }^q\left(2\right)}{{\mathcal{K}}_1(s\mid q)}\right)}^2|}}}.$$

By subtracting Equations (27) and (29), we obtain the following:

$$4{\mathbb{J}}_{\gamma }^q\left(3\right)\left(a_2-a_2^2\right)={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}({\mathsf{p}}_2-{\mathsf{v}}_2).$$

(34)

Performing some straightforward algebraic manipulations, we get the following:

$$a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}({\mathsf{p}}_2-{\mathsf{v}}_2)+a_2^2.$$

(35)

Substituting the value of $a_2^2$ from Equation (31), we obtain the following:

$$a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}({\mathsf{p}}_2-{\mathsf{v}}_2)+{\left({\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\right)}^2{\displaystyle \frac{{\mathsf{p}}_1^2+{\mathsf{v}}_1^2}{2{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2}}.$$

(36)

Finally, by applying Lemma 1 together with the triangle inequality, we arrive at the following upper bound:

$$|a_2|\le {\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2{\mathbb{J}}_{\gamma }^q\left(3\right)}}+{\left({\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{{\mathbb{J}}_{\gamma }^q\left(2\right)}}\right)}^2.$$

(37)

□

Theorem 2. 

Assume that $0<q<1$; the order γ is non-integer, For $f\in \aleph$ of the form (1) belonging to ${\mathbb{S}}_{\aleph }(s,q,\gamma )$ with $s\in (0.5, 1)$, the coefficients satisfy the following bound:

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\mathcal{K}}_1(s\mid q)\left|{\displaystyle \frac{1}{4{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right|,\hfill & if|\ell \left(\eta \right)|\le \left|{\displaystyle \frac{1}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right|,\hfill \\ 2{\mathcal{K}}_1(s\mid q)|\ell \left(\eta \right)|,\hfill & if|\ell \left(\eta \right)|\ge \left|{\displaystyle \frac{1}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right|,\hfill \end{array}\right.$$

(38)

where

$$\ell \left(\eta \right)={\displaystyle \frac{1-\eta }{4\left[2{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2-\left({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)\right){\left(\frac{{\mathbb{J}}_{\gamma }^q\left(2\right)}{{\mathcal{K}}_1(s\mid q)}\right)}^2\right]}}$$

and

$${\mathbb{J}}_{\gamma }^q\left(2\right)={\displaystyle \frac{{[1+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[1\right]}_q!}},{\mathbb{J}}_{\gamma }^q\left(3\right)={\displaystyle \frac{{[2+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[2\right]}_q!}}.$$

Proof. 

By referring to Equation (35), we have the following:

$$a_2-\eta a_2^2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}({\mathsf{p}}_2-{\mathsf{v}}_2)+(1-\eta )a_2^2.$$

(39)

Substituting the value of $a_2^2$ from the corresponding Equation (33), we get the following:

$$a_3-\eta a_2^2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)({\mathsf{p}}_2-{\mathsf{v}}_2)}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}+{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)(1-\eta )({\mathsf{p}}_2+{\mathsf{v}}_2)}{4\left[2{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2-({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)){\left(\frac{{\mathbb{J}}_{\gamma }^q\left(2\right)}{{\mathcal{K}}_1(s\mid q)}\right)}^2\right]}}.$$

(40)

After some simplifications and algebraic manipulations, we arrive at

$$a_2-\eta a_2^2={\mathcal{K}}_1(s\mid q)\left[\left(\ell \left(\eta \right)+{\displaystyle \frac{1}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right){\mathsf{p}}_2+\left(\ell \left(\eta \right)-{\displaystyle \frac{1}{8{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right){\mathsf{v}}_2\right],$$

where

$$\ell \left(\eta \right)={\displaystyle \frac{1-\eta }{4\left[2{\mathbb{J}}_{\gamma }^q\left(3\right)-{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2-\left({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)\right){\left(\frac{{\mathbb{J}}_{\gamma }^q\left(2\right)}{{\mathcal{K}}_1(s\mid q)}\right)}^2\right]}}.$$

□

Corollary 1. 

If $q\to 1$, then Theorem 1 reduces to the corresponding estimates for the classical Ruscheweyh operator ${\mathcal{R}}_{\gamma }$. In this limit, we set

$${\mathbb{J}}_{\gamma }\left(2\right)={\displaystyle \frac{(1+\gamma )!}{\gamma !1!}}=1+\gamma ,{\mathbb{J}}_{\gamma }\left(3\right)={\displaystyle \frac{(2+\gamma )!}{\gamma !2!}}={\displaystyle \frac{(\gamma +2)(\gamma +1)}{2}},$$

and we denote ${\mathcal{K}}_j\left(s\right):={\mathcal{K}}_j(s\mid 1)$$(j=1,2)$. Then the coefficient bounds become

$$|a_2|\le \sqrt{{\displaystyle \frac{{\mathcal{K}}_1\left(s\right)}{{\displaystyle \left|2{\mathbb{J}}_{\gamma }\left(3\right)-{\left[{\mathbb{J}}_{\gamma }\left(2\right)\right]}^2-\left({\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right)\right){\left(\frac{{\mathbb{J}}_{\gamma }\left(2\right)}{{\mathcal{K}}_1\left(s\right)}\right)}^2\right|}}}}$$

and

$$|a_3|\le {\displaystyle \frac{{\mathcal{K}}_1\left(s\right)}{2{\mathbb{J}}_{\gamma }\left(3\right)}}+{\left({\displaystyle \frac{{\mathcal{K}}_1\left(s\right)}{{\mathbb{J}}_{\gamma }\left(2\right)}}\right)}^2.$$

Moreover, the Fekete–Szegö inequality in this case reads

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\mathcal{K}}_1\left(s\right)\left|{\displaystyle \frac{1}{4{\mathbb{J}}_{\gamma }\left(3\right)}}\right|,\hfill & if|{\ell }_1\left(\eta \right)|\le \left|{\displaystyle \frac{1}{8{\mathbb{J}}_{\gamma }\left(3\right)}}\right|,\hfill \\ 2{\mathcal{K}}_1\left(s\right)\left|{\ell }_1\left(\eta \right)\right|,\hfill & if|{\ell }_1\left(\eta \right)|\ge \left|{\displaystyle \frac{1}{8{\mathbb{J}}_{\gamma }\left(3\right)}}\right|,\hfill \end{array}\right.$$

where

$${\ell }_1\left(\eta \right)={\displaystyle \frac{1-\eta }{4\left[2{\mathbb{J}}_{\gamma }\left(3\right)-{\left({\mathbb{J}}_{\gamma }\left(2\right)\right)}^2-\left({\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right)\right){\left(\frac{{\mathbb{J}}_{\gamma }\left(2\right)}{{\mathcal{K}}_1\left(s\right)}\right)}^2\right]}}.$$

Corollary 2. 

If $\gamma =0$ and $q\to 1$, then ${\mathcal{R}}_0^1$ is the identity and Theorem 1 reduces to estimates for the classical bi-starlike class ${\mathbb{S}}_{\aleph }\left(s\right)$. In this case,

$${\mathbb{J}}_0\left(2\right)=1,{\mathbb{J}}_0\left(3\right)=1,{\mathcal{K}}_j\left(s\right):={\mathcal{K}}_j(s\mid 1).$$

Hence,

$$|a_2|\le \sqrt{{\displaystyle \frac{{\mathcal{K}}_1\left(s\right)}{\left|1-{\displaystyle \frac{{\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right)}{{\mathcal{K}}_1{\left(s\right)}^2}}\right|}}}$$

and

$$|a_3|\le {\displaystyle \frac{{\mathcal{K}}_1\left(s\right)}{2}}+{\left({\mathcal{K}}_1\left(s\right)\right)}^2.$$

The Fekete–Szegö inequality becomes the following:

$$|a_3-\eta a_2^2|\le \left\{\begin{array}{cc}{\mathcal{K}}_1\left(s\right){\displaystyle \frac{1}{4}},\hfill & if|{\ell }_0\left(\eta \right)|\le {\displaystyle \frac{1}{8}},\hfill \\ 2{\mathcal{K}}_1\left(s\right)\left|{\ell }_0\left(\eta \right)\right|,\hfill & if|{\ell }_0\left(\eta \right)|\ge {\displaystyle \frac{1}{8}}\hfill \end{array}\right.$$

where

$${\ell }_0\left(\eta \right)={\displaystyle \frac{1-\eta }{4\left[1-{\displaystyle \frac{{\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right)}{{\mathcal{K}}_1{\left(s\right)}^2}}\right]}}.$$

3. Bounds for Initial Coefficients and Fekete–Szegö Inequalities in the Class ${\mathbb{K}}_{\aleph }(\mathit{s},\mathit{q},\mathit{\gamma })$

In this section, we introduce the convex class of bi-univalent functions defined via the q-Ruscheweyh operator and subordinated to q-Hermite polynomials. We then obtain sharp estimates for the initial coefficients of functions in this class and establish the corresponding Fekete–Szegö inequalities. These results shed light on the geometric structure and coefficient behavior of the newly introduced convex subclasses.

Definition 5. 

A function $f\in \aleph$ of the form (1) belongs to the convex class ${\mathbb{K}}_{\aleph }(s,q,\gamma )$ if

$$1+{\displaystyle \frac{\zeta {\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime \prime }}{{\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime }}}\prec \mathcal{M}(\zeta ,s,q),$$

(41)

and

$$1+{\displaystyle \frac{\mathsf{w}{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime \prime }}{{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime }}}\prec \mathcal{M}(\mathsf{w},s,q),$$

(42)

where g is the inverse function of f.

Theorem 3. 

Assume that $0<q<1$; the order γ is non-integer and $s\in (0.5, 1)$. For a function $f\in \aleph$ of the form (1) belongs to the class ${\mathbb{K}}_{\aleph }(s,q,\gamma )$. Then the following coefficient bounds hold:

$$|a_2|\le \sqrt{{\displaystyle \frac{2{\left({\mathcal{K}}_1(s\mid q)\right)}^3}{{\left({\mathcal{K}}_1(s\mid q)\right)}^2\left[12{\mathbb{J}}_{\gamma }^q\left(3\right)-8{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\right]-8{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\left({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)\right)}}}$$

(43)

and

$$|a_3|\le {\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{6{\mathbb{J}}_{\gamma }^q\left(3\right)}}+{\displaystyle \frac{{\left({\mathcal{K}}_1(s\mid q)\right)}^2}{4{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2}},$$

(44)

where

$${\mathbb{J}}_{\gamma }^q\left(2\right)={\displaystyle \frac{{[1+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[1\right]}_q!}},{\mathbb{J}}_{\gamma }^q\left(3\right)={\displaystyle \frac{{[2+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[2\right]}_q!}}.$$

Proof. 

Let $f\in \aleph$ be defined by (1) with $f\in {\mathbb{K}}_{\aleph }(s,q,\gamma )$. We consider

$$1+{\displaystyle \frac{\zeta {\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime \prime }}{{\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime }}}=\mathcal{M}(\beth \left(\zeta \right),s,q),1+{\displaystyle \frac{\mathsf{w}{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime \prime }}{{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime }}}=\mathcal{M}(\daleth \left(\mathsf{w}\right),s,q),$$

(45)

where $g=f^{-1}$ and $\beth \left(\zeta \right),\daleth \left(\mathsf{w}\right)\in \mathbb{D}$ are analytic functions as defined in Section 2.

Using the generating function $\mathcal{M}(\zeta ,s,q)$ from (9), we obtain the desired relations for coefficient estimates.

$$\begin{array}{cc}\hfill \mathcal{M}(\beth (\zeta ),s,q)& =1+{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{p}}_{1}z+\left[{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{p}}_2-{\displaystyle \frac{{\mathsf{p}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{p}}_1^2\right]{\zeta }^2+\dots ,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \mathcal{M}(\daleth (\mathsf{w}),s,q)& =1+{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{v}}_1\mathsf{w}+\left[{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{v}}_2-{\displaystyle \frac{{\mathsf{v}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{v}}_1^2\right]{\mathsf{w}}^2+\dots .\hfill \end{array}$$

(47)

On the other hand, applying the q-operator ${\mathcal{R}}_{\gamma }^q$ to f and g yields the following series expansions:

$$1+{\displaystyle \frac{\zeta {\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime \prime }}{{\left({\mathcal{R}}_{\gamma }^{q}f\left(\zeta \right)\right)}^{\prime }}}=1+2{\mathbb{J}}_{\gamma }^q\left(2\right)a_2\zeta +\left(-4{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2{a}_2^2+6{\mathbb{J}}_{\gamma }^q\left(3\right)a_3\right){\zeta }^2+\dots ,$$

(48)

and

$$1+{\displaystyle \frac{\mathsf{w}{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime \prime }}{{\left({\mathcal{R}}_{\gamma }^{q}g\left(\mathsf{w}\right)\right)}^{\prime }}}=1-2{\mathbb{J}}_{\gamma }^q\left(2\right)a_2\mathsf{w}+\left((12{\mathbb{J}}_{\gamma }^q\left(3\right)-4{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2)a_2^2-6{\mathbb{J}}_{\gamma }^q\left(3\right)a_3\right){\mathsf{w}}^2+\dots .$$

(49)

A comparison of the coefficients of $\zeta$ in Equations (46) and (48), together with the coefficients of $\mathsf{w}$ in Equations (47) and (49)), yields the following:

$$2{\mathbb{J}}_{\gamma }^q\left(2\right)a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{p}}_1,$$

(50)

$$6{\mathbb{J}}_{\gamma }^q\left(3\right)a_3-4{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2{a}_2^2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{p}}_2-{\displaystyle \frac{{\mathsf{p}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{p}}_1^2,$$

(51)

$$-2{\mathbb{J}}_{\gamma }^q\left(2\right)a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}{\mathsf{v}}_1,$$

(52)

and

$$(12{\mathbb{J}}_{\gamma }^q\left(3\right)-4{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2)a_2^2-6{\mathbb{J}}_{\gamma }^q\left(3\right)a_3={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\left({\mathsf{v}}_2-{\displaystyle \frac{{\mathsf{v}}_1^2}{2}}\right)+{\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}{\mathsf{v}}_1^2.$$

(53)

By referring to Equations (50) and (52), we obtain

$${\mathsf{p}}_1=-{\mathsf{v}}_1$$

(54)

and

$$8{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2{a}_2^2={\left({\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}\right)}^2\left({\mathsf{p}}_1^2+{\mathsf{v}}_1^2\right)⟹a_2^2={\displaystyle \frac{{\left[{\mathcal{K}}_1(s\mid q)\right]}^2}{32{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2}}({\mathsf{p}}_1^2+{\mathsf{v}}_1^2).$$

(55)

By adding Equations (51) and (53), we obtain

$$(12{\mathbb{J}}_{\gamma }^q\left(3\right)-8{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2)a_2^2=\left({\displaystyle \frac{{\mathcal{K}}_2(s\mid q)}{4}}-{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{4}}\right)({\mathsf{p}}_1^2+{\mathsf{v}}_1^2)+{\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}({\mathsf{p}}_2+{\mathsf{v}}_2).$$

(56)

Using Equation (55), we obtain

$$a_2^2={\displaystyle \frac{{\left(\frac{{\mathcal{K}}_1(s\mid q)}{2}\right)}^3({\mathsf{p}}_2+{\mathsf{v}}_2)}{{\left(\frac{{\mathcal{K}}_1(s\mid q)}{2}\right)}^2\left[12{\mathbb{J}}_{\gamma }^q\left(3\right)-8{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\right]-8{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\left(\frac{{\mathcal{K}}_2(s\mid q)}{4}-\frac{{\mathcal{K}}_1(s\mid q)}{4}\right)}}.$$

(57)

Now, we determine an upper bound for $|a_2|$. By applying Lemma 1, we obtain

$$|a_2|\le \sqrt{{\displaystyle \frac{2{\left({\mathcal{K}}_1(s\mid q)\right)}^3}{{\left({\mathcal{K}}_1(s\mid q)\right)}^2\left[12{\mathbb{J}}_{\gamma }^q\left(3\right)-8{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\right]-8{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\left({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q)\right)}}}.$$

By subtracting Equations (51) and (53), we obtain

$$12{\mathbb{J}}_{\gamma }^q\left(3\right)(a_3-a_2^2)={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{2}}({\mathsf{p}}_2-{\mathsf{v}}_2).$$

(58)

Performing some straightforward algebraic manipulations, we get

$$a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}({\mathsf{p}}_2-{\mathsf{v}}_2)+a_2^2.$$

(59)

Substituting the value of $a_2^2$ from Equation (55), we obtain

$$a_2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}({\mathsf{p}}_2-{\mathsf{v}}_2)+{\displaystyle \frac{{\left({\mathcal{K}}_1(s\mid q)\right)}^2}{32{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2}}({\mathsf{p}}_1^2+{\mathsf{v}}_1^2).$$

(60)

Finally, by applying Lemma 1 together with the triangle inequality, we arrive at the following upper bound:

$$|a_3|\le {\displaystyle \frac{4{\mathcal{K}}_1(s\mid q)}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}+{\displaystyle \frac{8{\left({\mathcal{K}}_1(s\mid q)\right)}^2}{32{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2}}={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{6{\mathbb{J}}_{\gamma }^q\left(3\right)}}+{\displaystyle \frac{{\left({\mathcal{K}}_1(s\mid q)\right)}^2}{4{\left[{\mathbb{J}}_{\gamma }^q\left(2\right)\right]}^2}}.$$

□

Theorem 4. 

Assume that $0<q<1$, the order γ is non-integer, and $s\in (0.5, 1)$. For $f\in \aleph$ of the form (1) and belong to the class ${\mathbb{K}}_{\aleph }(s,q,\gamma )$. Then the coefficients satisfy the following bound:

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\mathcal{K}}_1(s\mid q)\left|{\displaystyle \frac{1}{12{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right|,\hfill & if|\ell \left(\vartheta \right)|\le \left|{\displaystyle \frac{1}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right|,\hfill \\ 2{\mathcal{K}}_1(s\mid q)|\ell \left(\vartheta \right)|,\hfill & if|\ell \left(\vartheta \right)|\ge \left|{\displaystyle \frac{1}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right|,\hfill \end{array}\right.$$

(61)

where

$$\ell \left(\vartheta \right)={\displaystyle \frac{{\left({\mathcal{K}}_1(s\mid q)\right)}^2(1-\vartheta )}{8\left[{\left({\mathcal{K}}_1(s\mid q)\right)}^2\left(3{\mathbb{J}}_{\gamma }^q\left(3\right)-2{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\right)-2{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q))\right]}}$$

and

$${\mathbb{J}}_{\gamma }^q\left(2\right)={\displaystyle \frac{{[1+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[1\right]}_q!}},{\mathbb{J}}_{\gamma }^q\left(3\right)={\displaystyle \frac{{[2+\gamma ]}_q!}{{\left[\gamma \right]}_q!{\left[2\right]}_q!}}.$$

Proof. 

By referring to Equation (59), we have the following:

$$a_2-\vartheta a_2^2={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}({\mathsf{p}}_2-{\mathsf{v}}_2)+(1-\vartheta )a_2^2.$$

(62)

Substituting the value of $a_2^2$ from the corresponding Equation (57), we get the following:

$$\begin{array}{cc}\hfill a_3-\vartheta a_2^2& ={\displaystyle \frac{{\mathcal{K}}_1(s\mid q)({\mathsf{p}}_2-{\mathsf{v}}_2)}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}\hfill \\ \hfill & +{\displaystyle \frac{{\left({\mathcal{K}}_1(s\mid q)\right)}^3(1-\vartheta )({\mathsf{p}}_2+{\mathsf{v}}_2)}{8\left[{\left({\mathcal{K}}_1(s\mid q)\right)}^2\left(3{\mathbb{J}}_{\gamma }^q\left(3\right)-2{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\right)-2{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q))\right]}}.\hfill \end{array}$$

(63)

After some simplifications and algebraic manipulations, we arrive at

$$a_2-\vartheta a_2^2={\mathcal{K}}_1(s\mid q)\left[\left(\ell \left(\vartheta \right)+{\displaystyle \frac{1}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right){\mathsf{p}}_2+\left(\ell \left(\vartheta \right)-{\displaystyle \frac{1}{24{\mathbb{J}}_{\gamma }^q\left(3\right)}}\right){\mathsf{v}}_2\right],$$

where

$$\ell \left(\vartheta \right)={\displaystyle \frac{{\left({\mathcal{K}}_1(s\mid q)\right)}^2(1-\vartheta )}{8\left[{\left({\mathcal{K}}_1(s\mid q)\right)}^2\left(3{\mathbb{J}}_{\gamma }^q\left(3\right)-2{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2\right)-2{\left({\mathbb{J}}_{\gamma }^q\left(2\right)\right)}^2({\mathcal{K}}_2(s\mid q)-{\mathcal{K}}_1(s\mid q))\right]}}.$$

□

Corollary 3. 

If $q\to 1$, then the unified Theorem reduces to the corresponding result involving the classical Ruscheweyh differential operator ${\mathcal{R}}_{\gamma }$. In this case,

$${\mathbb{J}}_{\gamma }\left(2\right)=1+\gamma ,{\mathbb{J}}_{\gamma }\left(3\right)={\displaystyle \frac{(\gamma +2)(\gamma +1)}{2}},$$

and we set ${\mathcal{K}}_j\left(s\right):={\mathcal{K}}_j(s\mid 1)$ for $j=1,2$. Then the following sharp bounds hold:

$$|a_2|\le \sqrt{{\displaystyle \frac{2{\left({\mathcal{K}}_1\left(s\right)\right)}^3}{{\left({\mathcal{K}}_1\left(s\right)\right)}^2\left[12{\mathbb{J}}_{\gamma }\left(3\right)-8{\left({\mathbb{J}}_{\gamma }\left(2\right)\right)}^2\right]-8{\left({\mathbb{J}}_{\gamma }\left(2\right)\right)}^2\left({\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right)\right)}}},$$

$$|a_3|\le {\displaystyle \frac{{\mathcal{K}}_1\left(s\right)}{6{\mathbb{J}}_{\gamma }\left(3\right)}}+{\displaystyle \frac{{\left({\mathcal{K}}_1\left(s\right)\right)}^2}{4{\left[{\mathbb{J}}_{\gamma }\left(2\right)\right]}^2}}.$$

Moreover, the Fekete–Szegö functional satisfies the following:

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\mathcal{K}}_1\left(s\right)\left|{\displaystyle \frac{1}{12{\mathbb{J}}_{\gamma }\left(3\right)}}\right|,\hfill & if|{\ell }_1\left(\vartheta \right)|\le \left|{\displaystyle \frac{1}{24{\mathbb{J}}_{\gamma }\left(3\right)}}\right|,\hfill \\ 2{\mathcal{K}}_1\left(s\right)\left|{\ell }_1\left(\vartheta \right)\right|,\hfill & if|{\ell }_1\left(\vartheta \right)|\ge \left|{\displaystyle \frac{1}{24{\mathbb{J}}_{\gamma }\left(3\right)}}\right|,\hfill \end{array}\right.$$

where

$${\ell }_1\left(\vartheta \right)={\displaystyle \frac{{\left({\mathcal{K}}_1\left(s\right)\right)}^2(1-\vartheta )}{8\left[{\left({\mathcal{K}}_1\left(s\right)\right)}^2\left(3{\mathbb{J}}_{\gamma }\left(3\right)-2{\left({\mathbb{J}}_{\gamma }\left(2\right)\right)}^2\right)-2{\left({\mathbb{J}}_{\gamma }\left(2\right)\right)}^2\left({\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right)\right)\right]}}.$$

Corollary 4. 

If $\gamma =0$ and $q\to 1$, then the operator ${\mathcal{R}}_0^1$ becomes the identity and the unified Theorem reduces to the coefficient estimates for the classical bi-starlike class ${\mathbb{K}}_{\aleph }\left(s\right)$. In this case,

$${\mathbb{J}}_0\left(2\right)=1,{\mathbb{J}}_0\left(3\right)=1,{\mathcal{K}}_j\left(s\right):={\mathcal{K}}_j(s\mid 1).$$

Thus, the bounds take the form

$$|a_2|\le \sqrt{{\displaystyle \frac{2{\left({\mathcal{K}}_1\left(s\right)\right)}^3}{4{\left({\mathcal{K}}_1\left(s\right)\right)}^2-8({\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right))}}},$$

$$|a_3|\le {\displaystyle \frac{{\mathcal{K}}_1\left(s\right)}{6}}+{\displaystyle \frac{{\left({\mathcal{K}}_1\left(s\right)\right)}^2}{4}}.$$

Moreover, the Fekete–Szegö inequality becomes the following:

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\mathcal{K}}_1\left(s\right){\displaystyle \frac{1}{12}},\hfill & if|{\ell }_0\left(\vartheta \right)|\le {\displaystyle \frac{1}{24}},\hfill \\ 2{\mathcal{K}}_1\left(s\right)\left|{\ell }_0\left(\vartheta \right)\right|,\hfill & if|{\ell }_0\left(\vartheta \right)|\ge {\displaystyle \frac{1}{24}},\hfill \end{array}\right.$$

where

$${\ell }_0\left(\vartheta \right)={\displaystyle \frac{{\left({\mathcal{K}}_1\left(s\right)\right)}^2(1-\vartheta )}{8\left[{\left({\mathcal{K}}_1\left(s\right)\right)}^2-2({\mathcal{K}}_2\left(s\right)-{\mathcal{K}}_1\left(s\right))\right]}}.$$

Remark 5. 

The approach developed here remains valid for other q-orthogonal polynomial families, such as q-Chebyshev polynomials, provided the associated generating functions are analytic, normalized, and possess a positive real part in the unit disk.

4. Numerical Illustrations

In order to demonstrate the applicability of the obtained coefficient bounds beyond the limiting cases $q\to 1$ and $\gamma =0$, we now provide explicit numerical illustrations for admissible parameter values.

Example 4. 

Let

$$q=0.5,s=0.7,\gamma =\frac{1}{2}.$$

Then

$${\left[\frac{3}{2}\right]}_{0.5}={\displaystyle \frac{1-{0.5}^{1.5}}{1-0.5}}={\displaystyle \frac{1-0.3536}{0.5}}\approx 1.2928.$$

If

$$f\left(\zeta \right)=\zeta +a_2{\zeta }^2+a_3{\zeta }^3+\dots \in {\mathbb{S}}_{\aleph }(s,q,\gamma ),$$

then by Theorem 1, the coefficient $a_2$ satisfies

$$|a_2|\le 1.083.$$

This provides an explicit numerical upper bound for $|a_2|$ corresponding to fractional order $\gamma =\frac{1}{2}$.

Example 5. 

Let

$$q=0.8,s=0.7,\gamma =\frac{1}{2}.$$

Then

$${\left[\frac{3}{2}\right]}_{0.8}={\displaystyle \frac{1-{0.8}^{1.5}}{1-0.8}}={\displaystyle \frac{1-0.7155}{0.2}}\approx 1.4225.$$

If $f\in {\mathbb{S}}_{\aleph }(s,q,\gamma )$, then again by Theorem 1,

$$|a_2|\le 0.984.$$

This shows that increasing the parameter q leads to a tighter bound on the second coefficient.

Example 6. 

Let

$$q=0.5,s=0.7,\gamma =\frac{1}{2},\eta =1.$$

For these values,

$${[\gamma +1]}_q={\displaystyle \frac{1-{0.5}^{1.5}}{1-0.5}}\approx 1.2928.$$

Assume that

$$f\left(\zeta \right)=\zeta +a_2{\zeta }^2+a_3{\zeta }^3+\dots \in {\mathbb{S}}_{\aleph }(s,q,\gamma ).$$

By Theorem 2, the Fekete–Szegö functional satisfies

$$|a_3-a_2^2|\le 0.837.$$

This numerical value illustrates the effectiveness of the Fekete–Szegö inequality for fractional order $\gamma =\frac{1}{2}$ and intermediate values of $q\in (0, 1)$.

Example 7. 

Let

$$q=0.8,s=0.7,\gamma =\frac{1}{2}.$$

Then

$${[\gamma +1]}_q={\displaystyle \frac{1-{0.8}^{1.5}}{1-0.8}}\approx 1.4225.$$

Suppose that

$$f\left(\zeta \right)=\zeta +a_2{\zeta }^2+a_3{\zeta }^3+\dots \in {\mathbb{K}}_{\aleph }(s,q,\gamma ).$$

According to Theorem 3, the second coefficient satisfies

$$|a_2|\le 0.492.$$

This provides a concrete numerical bound for functions in the bi-convex subclass associated with the fractional q-Ruscheweyh operator.

Example 8. 

Let

$$q=0.5,s=0.7,\gamma =\frac{1}{2},\eta =1.$$

Assume that

$$f\in {\mathbb{K}}_{\aleph }(s,q,\gamma ).$$

By Theorem 4, the Fekete–Szegö functional satisfies

$$|a_3-a_2^2|\le 0.418.$$

This numerical bound demonstrates that the bi-convex subclass yields smaller Fekete–Szegö bounds compared to the corresponding bi-starlike class.

Remark 6. 

The above numerical examples confirm that the coefficient bounds and Fekete–Szegö inequalities obtained in Theorems 1–4 remain effective for fractional orders $\gamma \in (0, 1)$ and intermediate values of $q\in (0, 1)$. In particular, the bounds become tighter as q increases, which is consistent with the limiting behavior as $q\to 1$.

5. Conclusions

In this paper, we introduced new subclasses of bi-univalent functions defined via the fractional q-Ruscheweyh operator and subordinated to q-Hermite polynomials. We obtained sharp bounds for the first few Taylor-Maclaurin coefficients and established Fekete–Szegö inequalities, providing clear relations between the second and third coefficients. The results also include several special cases, such as taking $q\to 1$ or $\gamma =0$, which recover known bounds for classical Ruscheweyh operators and bi-starlike functions. This shows that our findings not only generalize previous results but also offer a flexible framework for studying other subclasses of bi-univalent functions. Overall, this work highlights the usefulness of q-calculus in geometric function theory and opens the door for further studies on coefficient problems and operator-based subclasses in analytic function theory.