Initial Coefficient Bounds for Bi-Close-to-Convex and Bi-Quasi-Convex Functions with Bounded Boundary Rotation Associated with q-Sălăgean Operator

Abstract

In this article, through the application of the q-Sălăgean operator associated with functions characterized by bounded boundary rotation, we propose a few new subclasses of bi-univalent functions that utilize the q-Sălăgean operator with bounded boundary rotation in the open unit disk $\mathbb{E}$. For these classes, we establish the initial bounds for the coefficients $|a_2|$ and $|a_3|$. Additionally, we have derived the well-known Fekete–Szegö inequality for this newly defined subclasses.

1. Introduction and Definitions

Let $\mathcal{A}$ represent the collection of functions h that are analytic within the open unit disk $\mathbb{E}=\{\nu \in \mathbb{C}:|\nu |<1\}$, expressed in the form

$$h\left(\nu \right)=\nu +\sum _{m=2}^{\infty }{a}_m{\nu }^m,\nu \in \mathbb{E}.$$

(1)

Additionally, let $\mathcal{S}$ represent the subclass of $\mathcal{A}$ that includes functions of the form (1), which are univalent in the domain $\mathbb{E}$. A significant and well-studied subclass of $\mathcal{S}$ is the class ${\mathcal{S}}^{*}\left(\mu \right)$ of starlike functions of order $\mu$ (where $0\le \mu <1)$, characterized by the condition

$$\Re \left({\displaystyle \frac{\nu h^{\prime }\left(\nu \right)}{h\left(\nu \right)}}\right)>\mu ,$$

for all $\nu$ in $\mathbb{E}$. Similarly, the class $\mathcal{C}\left(\mu \right)$ of convex functions of order $\mu$$(where0\le \mu <1)$ is defined by the condition

$$\Re \left(1+{\displaystyle \frac{\nu h^{\prime \prime }\left(\nu \right)}{h^{\prime }\left(\nu \right)}}\right)>\mu ,$$

for all $\nu$ in $\mathbb{E}$. Particularly, for $\mu =0$, the aforementioned classes correspond to the established classes ${\mathcal{S}}^{*}$ and $\mathcal{C}$, which represent the class of starlike functions and the class of convex functions, respectively. For the basic concepts about analytic univalent functions, interested readers may refer to [1]. Note that every function $h\in \mathcal{S}$ has an inverse $h^{-1}$, as defined by the Koebe One-Quarter Theorem (see [1]), which states that

$$h^{-1}\left(h\left(\nu \right)\right)=\nu ,\nu \in \mathbb{E}$$

and

$$h\left(h^{-1}\left(u\right)\right)=u,\left(|u|<r_0\left(h\right);r_0\left(h\right)\ge {\displaystyle \frac{1}{4}}\right),$$

holds true. Here, $h^{-1}\left(u\right)=\varphi \left(u\right)$ is defined as

$$h^{-1}\left(u\right)=\varphi \left(u\right)=u-a_2{u}^2+(2a_2^2-a_3)u^3-(5a_2^3-5a_2{a}_3+a_4)u^4+\dots .$$

(2)

A function h is classified as bi-univalent in the domain $\mathbb{E}$ if both h and its inverse $h^{-1}$ are univalent within $\mathbb{E}$. Let $\sigma$ represent the class of bi-univalent functions in $\mathbb{E}$ as defined by (2). The inequality $|a_2|\le 1.51$ is derived from Lewin [2]. Brannan and Taha [3] calculated the values of $|a_2|$ and $|a_3|$ for functions within the classes ${\mathcal{S}}_{\sigma }^{*}\left(\mu \right)$ and ${\mathcal{C}}_{\sigma }^{*}\left(\mu \right)$ drawing inspiration from Lewin [2], who investigated a class of bi-starlike and bi-convex functions. Additionally, Brannan and Clunie [4] conjectured that $|a_2|\le \sqrt{2}$, which seems partially true for few subclasses of $\sigma$ (see [5]). Note that function ${\displaystyle h\left(\nu \right)=\frac{\nu }{1-\nu }}$ is bi-convex. However, the Koebe function is not bi-univalent. Research on bi-univalent functions after 2010 was primarily popularized by the work of Srivastava et al. [6], who investigated several motivating subclasses of $\sigma$ and established the first two initial non-sharp estimates on Taylor–Maclaurin coefficients. Finding sharp estimates for the bi-univalent class $\sigma$ and establishing general coefficient for $|a_n|$ for $\sigma$ remains an open problem.

Let ${\mathcal{K}}_{\mu }$ denote the family of analytic functions $h\left(\nu \right)$ characterized by the form (1), with the constraint $0\le \mu \le 1$, and satisfying the condition

$$\left|arg\left({\displaystyle \frac{h^{\prime }\left(\nu \right)}{g^{\prime }\left(\nu \right)}}\right)\right|<{\displaystyle \frac{\mu \pi }{2}}$$

where g is a convex function.

Kaplan [7] and Reade [8] conducted an examination of these classes. Consequently, ${\mathcal{K}}_0$ is defined as $\mathcal{C}$, while ${\mathcal{K}}_1$ represents the families of convex univalent functions and close-to-convex functions, respectively. Additionally, ${\mathcal{K}}_{{\mu }_1}$ is considered a proper subclass of ${\mathcal{K}}_{{\mu }_2}$ when ${\mu }_1<{\mu }_2$. The class of close-to-convex functions of order $\mu$, where $0\le \mu \le 1$, as established in Reade’s work [8], is described as follows:

$$\Re \left({\displaystyle \frac{h^{\prime }\left(\nu \right)}{g^{\prime }\left(\nu \right)}}\right)>\mu .$$

Let $0\le \mu <1$. A function $h\in \mathcal{A}$, characterized by the form (1) is classified as belonging to the family of quasi-convex functions of order $\mu$ if there exists a function $g\in \mathcal{C}$ such that

$$\Re \left({\displaystyle \frac{h^{\prime }\left(\nu \right)+\nu h^{\prime \prime }\left(\nu \right)}{g\left(\nu \right)}}\right)>\mu .$$

The collection of all quasi-convex functions of order $\mu$ is denoted by ${\mathcal{Q}}^{*}\left(\mu \right)$. It is important to note that every quasi-convex function is also close-to-convex. A function $h\in {\mathcal{Q}}^{*}\left(\mu \right)$ is equivalent to $\nu h^{\prime }\in \mathcal{K}\left(\mu \right)$. For further information regarding quasi-convex functions, one may refer to the work of [9].

A function h is referred to as having bounded boundary rotation if the range of h exhibits bounded boundary rotation. It is important to remember that bounded boundary rotation is characterized by the total variation of the tangent’s angle to the boundary curve over a complete circuit. Let h map the domain $\mathcal{E}$ onto the domain $\mathfrak{G}$. If $\mathfrak{G}$ is a Schlicht domain with a continuously differentiable boundary curve, and $\pi \theta \left(t\right)$ represents the angle of the tangent vector at the point $h\left(e^{it}\right)$ relative to the positive real axis, then the boundary rotation of $\mathfrak{G}$ is given by $\pi {\int }_0^{2\pi }|d\theta \left(t\right)|$. For details on bounded boundary rotation, one may refer to the work of Pinchuk [10]. In cases where $\mathfrak{G}$ lacks a sufficiently smooth boundary curve, the boundary rotation is determined through a limiting process. In 1975, Padmanabhan and Parvatham [11] introduced the class ${\mathcal{P}}_m\left(\mu \right)$. A function q is classified within ${\mathcal{P}}_m\left(\mu \right)$ if it is normalized, such that $q\left(0\right)=1$ and $q^{\prime }\left(0\right)>1$, and if it meets the following condition:

$${\int }_0^{2\pi }\Re \left|{\displaystyle \frac{q\left(\nu \right)-\mu }{1-\mu }}\right|d\left(t\right)\le m\pi ,$$

where $\nu =r{e}^{it}\in \mathbb{E}$. The function q belongs to the class ${\mathcal{P}}_m\left(\mu \right)$ if and only if there exists a non-decreasing function $\theta$ defined on the interval $[0,2\pi ]$ such that

$${\int }_0^{2\pi }d\theta \left(t\right)=2and{\int }_0^{2\pi }d\theta \left(t\right)\le m,m\ge 2,$$

which satisfies the equation

$$q\left(\nu \right)={\int }_0^{2\pi }{\displaystyle \frac{1+(1-2\mu )\nu e^{-it}}{1-\nu e^{-it}}}d\theta \left(t\right).$$

If we select $\mu =0$, the class ${\mathcal{P}}_m\left(\mu \right)$ is simplified to the class ${\mathcal{P}}_m$, as originally defined by Pinchuk [10] and subsequently examined by Robertson [12]. By selecting $\mu =0$ and $m=2$, the class ${\mathcal{P}}_m\left(\mu \right)$ is simplified to the class $\mathcal{P}$, recognized as the class of Carathéodory functions.

The concept of q-calculus was first established by Jackson [13], who systematically developed the ideas of the q-integral and q-derivative. Further studies on quantum groups have revealed a geometric perspective on q-analysis, suggesting a relationship between integrable systems and q-analysis. A comprehensive review of the applications of q-calculus in operator theory is available in [14].

For values of $0<q<1$, the Jackson q-derivative of a function h that is part of the class $\mathcal{A}$ is defined in the following way:

$${\mathcal{D}}_{q}h\left(\nu \right)=\left\{\begin{array}{cc}{\displaystyle \frac{h\left(q\nu \right)-h\left(\nu \right)}{\nu (q-1)}}\hfill & if\nu \ne 0,\hfill \\ h^{\prime }\left(0\right)\hfill & if\nu =0,\hfill \end{array}\right.$$

(3)

and ${\mathcal{D}}_q^{2}h\left(\nu \right)=\mathcal{D}\left({\mathcal{D}}_{q}h\left(\nu \right)\right)$. Now, from (3), we have

$${\mathcal{D}}_{q}h\left(\nu \right)=1+\sum _{m=2}^{\infty }{\left[m\right]}_q{a}_m{\nu }^m,$$

where

$${\left[m\right]}_q={\displaystyle \frac{q^m-1}{q-1}}$$

is sometimes called the basic number m. If $q\to 1^{-}$, ${\left[m\right]}_q\to m$. For a function $g\left(z\right)=z^m$, we obtain ${\mathcal{D}}_{q}g\left(z\right)={\left[m\right]}_q{z}^{m-1}$ and ${lim}_{q\to 1^{-}}{\mathcal{D}}_{q}g\left(z\right)=m{z}^{m-1}=g^{\prime }\left(z\right)$. The Sălăgean q-differential operator [15] for functions h belonging to the class $\mathcal{A}$ is detailed below:

$$\begin{array}{cc}\hfill {\mathcal{D}}_q^{0}h\left(\nu \right)=& h\left(\nu \right),\hfill \\ \hfill {\mathcal{D}}_q^{1}h\left(\nu \right)=& \nu {\mathcal{D}}_{q}h\left(\nu \right),\hfill \\ \hfill {\mathcal{D}}_q^{k}h\left(\nu \right)=& \nu {\mathcal{D}}_q\left({\mathcal{D}}_q^{k-1}h\left(\nu \right)\right),\hfill \\ \hfill {\mathcal{D}}_q^{k}h\left(\nu \right)=& \nu +\sum _{m=2}^{\infty }{\left[m\right]}_q^k{a}_m{\nu }^m,k\in \mathbb{N}\cup \left\{0\right\}.\hfill \end{array}$$

We point out that if $q\to 1^{-}$,

$${\mathcal{D}}^{k}h\left(\nu \right)=\nu +\sum _{m=2}^{\infty }{m}^k{a}_m{\nu }^m,k\in \mathbb{N}\cup \left\{0\right\},$$

is a well-known Sălăgean derivative [16]. In this study, we utilize the Sălăgean q-differential operator to define ${\mathcal{D}}^k\varphi \left(u\right)$ for functions h of the form (2). Thus, we have

$${\mathcal{D}}^k\varphi \left(u\right)={\mathcal{D}}^k{h}^{-1}\left(u\right)=u-a_2{\left[2\right]}_q^k{u}^2+(2a_2^2-a_3){\left[3\right]}_q^k{u}^3-(5a_2^3-5a_2{a}_3+a_4){\left[4\right]}_q^k{u}^4+\dots .$$

(4)

The following lemmas are essential for establishing our main theorems:

Lemma 1 

([17]). If a function $t\in {\mathcal{P}}_m\left(\mu \right)$ is expressed as

$$t\left(\nu \right)=1+t_1\nu +t_2{\nu }^2+t_3{\nu }^3+\dots ,$$

then for every $m\ge 1$, it holds that

$$|t_m|\le m(1-\mu ).$$

This result is accurate.

Lemma 2 

([1]). If a function $g\in \mathcal{C}$ is expressed as

$$g\left(\nu \right)=\nu +b_2{\nu }^2+b_3{\nu }^3+\dots ,$$

then for every $m\ge 2$, it holds that

$$|b_m|\le 1.$$

This result is accurate.

Lemma 3 

([18]). If a function $g\in \mathcal{C}$ is expressed as

$$g\left(\nu \right)=\nu +b_2{\nu }^2+b_3{\nu }^3+\dots ,$$

then for every $\vartheta \in \mathbb{R}$, it holds that

$$|b_3-\vartheta b_2^2|\le \left\{\begin{array}{cc}1-\vartheta \hfill & {\displaystyle if\vartheta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{1}{3}}\hfill & {\displaystyle if\frac{2}{3}\le \vartheta \le \frac{4}{3},}\hfill \\ \vartheta -1\hfill & {\displaystyle if\vartheta >\frac{4}{3}.}\hfill \end{array}\right.$$

In this article, through the application of the q-Sălăgean operator associated with functions characterized by bounded boundary rotation, we propose a few new subclasses of bi-univalent functions that utilize the q-Sălăgean operator with bounded boundary rotation in the open unit disk $\mathbb{E}$. For these classes, we establish the initial bounds for the coefficients $|a_2|$ and $|a_3|$. Additionally, we have derived the well-known Fekete–Szegö inequality for these newly defined subclass functions.

2. Main Results

Definition 1. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. Let a function $h\in \mathcal{A}$ be defined by (1). We denote h as belonging to the class ${\mathcal{K}}_{\sigma }^k(m,\mu ,q)$ if there exist two convex functions, $g\left(\nu \right)$ and $\psi \left(u\right)$, that satisfy the conditions:

$${\displaystyle \frac{{\mathcal{D}}_q^{k+1}h\left(\nu \right)}{{\mathcal{D}}_q^{k+1}g\left(\nu \right)}}\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\displaystyle \frac{{\mathcal{D}}_q^{k+1}\varphi \left(u\right)}{{\mathcal{D}}_q^{k+1}\psi \left(u\right)}}\in {\mathcal{P}}_m\left(\mu \right).$$

Remark 1. 

1. By opting for $q\to 1^{-}$, the class ${\mathcal{K}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{K}}_{\sigma }^k(m,\mu )$. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$${\displaystyle \frac{{\left({\mathcal{D}}^{k}h\right)}^{\prime }\left(\nu \right)}{{\left({\mathcal{D}}^{k}g\right)}^{\prime }\left(\nu \right)}}\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\displaystyle \frac{{\left({\mathcal{D}}^k\varphi \right)}^{\prime }\left(u\right)}{{\left({\mathcal{D}}^k\psi \right)}^{\prime }\left(u\right)}}\in {\mathcal{P}}_m\left(\mu \right).$$

2. By opting for $q\to 1^{-}$ and $k=0$, the class ${\mathcal{K}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{K}}_{\sigma }(m,\mu )$, which was established by Breaz [19] in 2023. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$${\displaystyle \frac{h^{\prime }\left(\nu \right)}{g^{\prime }\left(\nu \right)}}\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\displaystyle \frac{{\varphi }^{\prime }\left(u\right)}{{\psi }^{\prime }\left(u\right)}}\in {\mathcal{P}}_m\left(\mu \right).$$

3. By opting for $q\to 1^{-}$, $k=0$ and $m=2$, the class ${\mathcal{K}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{K}}_{\sigma }\left(\mu \right)$, which was established by Sivasubramanian [5] in 2015. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$$\Re \left({\displaystyle \frac{h^{\prime }\left(\nu \right)}{g^{\prime }\left(\nu \right)}}\right)>\mu$$

and

$$\Re \left({\displaystyle \frac{{\varphi }^{\prime }\left(u\right)}{{\psi }^{\prime }\left(u\right)}}\right)>\mu .$$

Theorem 1. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{K}}_{\sigma }^k(m,\mu ,q)$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{3{\left[3\right]}_q^k}}},$$

(5)

$$|a_3|\le {\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{3{\left[3\right]}_q^k}}$$

(6)

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{(1-\vartheta )[3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)]}{3{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta <0,\hfill \\ {\displaystyle \frac{3{\left[3\right]}_q^k(1-\vartheta )+m(1-\mu )(2{\left[2\right]}_q^k(1-\vartheta )+1)}{3{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}0\le \vartheta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{{\left[3\right]}_q^k+m(1-\vartheta )(2{\left[2\right]}_q^k(1-\vartheta )+1)}{3{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \vartheta <1,}\hfill \\ {\displaystyle \frac{{\left[3\right]}_q^k+m(\vartheta -1)(2{\left[2\right]}_q^k(\vartheta -1)+1)}{3{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}1\le \vartheta \le \frac{4}{3},}\hfill \\ {\displaystyle \frac{3{\left[3\right]}_q^k(\vartheta -1)+m(1-\mu )(2{\left[2\right]}_q^k(\vartheta -1)+1)}{3{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}\frac{4}{3}<\vartheta <2,}\hfill \\ {\displaystyle \frac{(\vartheta -1)[3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)]}{3{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

(7)

Proof. 

Consider $h\in {\mathcal{K}}_{\sigma }^k(m,\mu ,q)$. Based on Definition 1, we can define two analytic functions, $t\left(\nu \right)$ and $y\left(u\right)$, such that $t\left(0\right)=y\left(0\right)=1$. These functions belong to the set ${\mathcal{P}}_m\left(\mu \right)$ and fulfill the following conditions:

$${\displaystyle \frac{{\mathcal{D}}_q^{k+1}h\left(\nu \right)}{{\mathcal{D}}_q^{k+1}g\left(\nu \right)}}=t\left(\nu \right)$$

(8)

and

$${\displaystyle \frac{{\mathcal{D}}_q^{k+1}\varphi \left(u\right)}{{\mathcal{D}}_q^{k+1}\psi \left(u\right)}}=y\left(u\right),$$

(9)

where

$$t\left(\nu \right)=1+t_1\nu +t_2{\nu }^2+t_3{\nu }^3+\dots$$

(10)

and

$$y\left(u\right)=1+y_{1}u+y_2{u}^2+y_3{u}^3+\dots .$$

(11)

Equations (8) and (9) may be reformulated as follows:

$${\mathcal{D}}_q^{k+1}h\left(\nu \right)={\mathcal{D}}_q^{k+1}g\left(\nu \right)t\left(\nu \right)$$

(12)

and

$${\mathcal{D}}_q^{k+1}\varphi \left(u\right)={\mathcal{D}}_q^{k+1}\psi \left(u\right)y\left(u\right).$$

(13)

Consequently, through the comparison of Equations (10)–(13), we obtain

$$2{\left[2\right]}_q^k{a}_2=2{\left[2\right]}_q^k{d}_2+t_1,$$

(14)

$$3{\left[3\right]}_q^k{a}_3=3{\left[3\right]}_q^k{d}_3+2{\left[2\right]}_q^k{a}_2{t}_1+t_2,$$

(15)

$$-2{\left[2\right]}_q^k{a}_2=-2{\left[2\right]}_q^k{d}_2+y_1$$

(16)

and

$$6{\left[3\right]}_q^k{a}_2^2-3{\left[3\right]}_q^k{a}_3=6{\left[3\right]}_q^k{d}_2^2-3{\left[3\right]}_q^k{d}_3-2{\left[2\right]}_q^k{a}_2{y}_1+y_2.$$

(17)

Therefore, by summing Equations (14) and (16), we obtain $t_1+y_1=0$. Additionally, by adding Equations (15) and (17) and utilizing the relationship $t_1+y_1=0$, we arrive at the following result:

$$6{\left[3\right]}_q^k{a}_2^2=6{\left[3\right]}_q^k{d}_2^2+4{\left[2\right]}_q^k{a}_2{t}_1+t_2+y_2.$$

(18)

Through the application of Lemmas 1 and 2 in Equation (18), we obtain the following outcome:

$$|a_2{|}^2\le {\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{3{\left[3\right]}_q^k}}.$$

(19)

According to Equation (19), the bound of $|a_2|$ is specified in Equation (5). Similarly, through the application of Lemmas 1 and 2 in Equation (15), we obtain the following outcome:

$$|a_3|\le {\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{3{\left[3\right]}_q^k}}.$$

(20)

According to Equation (20), the bound of $|a_3|$ is specified in Equation (6). Based on Equations (15) and (18), it follows that for any real number $\vartheta$, we obtain the result

$$a_3-\vartheta a_2^2={\displaystyle \frac{6{\left[3\right]}_q^k(d_3-\vartheta d_2^2)+4{\left[2\right]}_q^k{d}_2{t}_1(1-\vartheta )+t_2(2-\vartheta )-\vartheta y_2}{6{\left[3\right]}_q^k}}.$$

(21)

Through the application of Lemmas 1 and 2 in Equation (21), we obtain the following outcome:

$$|a_3-\vartheta a_2^2|\le {\displaystyle \frac{6{\left[3\right]}_q^k|d_3-\vartheta d_2^2{|+4\left[2\right]}_q^{k}m(1-\mu )|1-\vartheta |+m(1-\mu )[|2-\vartheta |+|\vartheta |]}{6{\left[3\right]}_q^k}}.$$

(22)

Employing Lemma 3 within the context of Equation (22) leads us to the bound of $|a_3-\vartheta a_2^2|$ specified in Equation (7). This signifies the end of the proof for Theorem 1. □

By opting for $q\to 1^{-}$ in Theorem 1, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{K}}_{\sigma }^k(m,\mu )$.

Corollary 1. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{K}}_{\sigma }^k(m,\mu )$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{3.3^k+m(1-\mu )(2.2^k+1)}{3.3^k}}},$$

$$|a_3|\le {\displaystyle \frac{3.3^k+m(1-\mu )(2.2^k+1)}{3.3^k}}$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{(1-\vartheta )[3.3^k+m(1-\mu )(2.2^k+1)]}{3.3^k}}\hfill & \mathit{for}\vartheta <0,\hfill \\ {\displaystyle \frac{3.3^k(1-\vartheta )+m(1-\mu )(2.2^k(1-\vartheta )+1)}{3.3^k}}\hfill & {\displaystyle \mathit{for}0\le \vartheta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{3^k+m(1-\vartheta )(2.2^k(1-\vartheta )+1)}{3.3^k}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \vartheta <1,}\hfill \\ {\displaystyle \frac{3^k+m(\vartheta -1)(2.2^k(\vartheta -1)+1)}{3.3^k}}\hfill & {\displaystyle \mathit{for}1\le \vartheta \le \frac{4}{3},}\hfill \\ {\displaystyle \frac{3.3^k(\vartheta -1)+m(1-\mu )(2.2^k(\vartheta -1)+1)}{3.3^k}}\hfill & {\displaystyle \mathit{for}\frac{4}{3}<\vartheta <2,}\hfill \\ {\displaystyle \frac{(\vartheta -1)[3.3^k+m(1-\mu )(2.2^k+1)]}{3.3^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

By opting for $q\to 1^{-}$ and $k=0$ in Theorem 1, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{K}}_{\sigma }(m,\mu )$.

Corollary 2. 

Assume that $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{K}}_{\sigma }(m,\mu )$, then

$$|a_2|\le \sqrt{1+m(1-\mu )},$$

$$|a_3|\le 1+m(1-\mu )$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}(1-\vartheta )[1+m(1-\mu \left)\right]\hfill & \mathit{for}\vartheta <0,\hfill \\ {\displaystyle \frac{3(1-\vartheta )+m(1-\mu )\left(2\right(1-\vartheta )+1)}{3}}\hfill & {\displaystyle \mathit{for}0\le \vartheta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{1+m(1-\vartheta )\left(2\right(1-\vartheta )+1)}{3}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \vartheta <1,}\hfill \\ {\displaystyle \frac{1+m(\vartheta -1)\left(2\right(\vartheta -1)+1)}{3}}\hfill & {\displaystyle \mathit{for}1\le \vartheta \le \frac{4}{3},}\hfill \\ {\displaystyle \frac{3(\vartheta -1)+m(1-\mu )\left(2\right(\vartheta -1)+1)}{3}}\hfill & {\displaystyle \mathit{for}\frac{4}{3}<\vartheta <2,}\hfill \\ (\vartheta -1)[1+m(1-\mu \left)\right]\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

The choice of the function $g\left(\nu \right)=\nu$ in Definition 1 leads to a significant reduction of the class ${\mathcal{K}}_{\sigma }^k(m,\mu ,q)$ to the class ${\mathcal{H}}_{\sigma }^k(m,\mu ,q)$. Functions h within the class ${\mathcal{H}}_{\sigma }^k(m,\mu ,q)$ are those that are part of $\sigma$ and meet the following specified requirements:

$${\mathcal{D}}_q^{k+1}h\left(z\right)\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\mathcal{D}}_q^{k+1}\varphi \left(w\right)\in {\mathcal{P}}_m\left(\mu \right).$$

Remark 2. 

1. By opting for $q\to 1^{-}$, the class ${\mathcal{H}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{H}}_{\sigma }^k(m,\mu )$. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$${\left({\mathcal{D}}^{k}h\right)}^{\prime }\left(\nu \right)\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\left({\mathcal{D}}^k\varphi \right)}^{\prime }\left(u\right)\in {\mathcal{P}}_m\left(\mu \right).$$

2. By opting for $q\to 1^{-}$ and $k=0$, the class ${\mathcal{H}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{H}}_{\sigma }(m,\mu )$, which was established by Li [20] in 2020. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$$h^{\prime }\left(\nu \right)\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\varphi }^{\prime }\left(u\right)\in {\mathcal{P}}_m\left(\mu \right).$$

3. By opting for $q\to 1^{-}$, $k=0$ and $m=2$, the class ${\mathcal{H}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{H}}_{\sigma }\left(\mu \right)$, which was established by Srivastava [6] in 2010. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$$\Re \left(h^{\prime }\left(\nu \right)\right)>\mu$$

and

$$\Re \left({\varphi }^{\prime }\left(u\right)\right)>\mu .$$

Theorem 2. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{H}}_{\sigma }^k(m,\mu ,q)$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{m(1-\mu )}{3{\left[3\right]}_q^k}}}$$

(23)

$$|a_3|\le {\displaystyle \frac{m(1-\mu )}{3{\left[3\right]}_q^k}}$$

(24)

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\mu )(1-\vartheta )}{3{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta \le 0,\hfill \\ {\displaystyle \frac{m(1-\mu )}{3{\left[3\right]}_q^k}}\hfill & \mathit{for}0\le \vartheta \le 2,\hfill \\ {\displaystyle \frac{m(1-\mu )(\vartheta -1)}{3{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

(25)

Proof. 

The proof of Theorem 2 is straightforward and hence chosen to omit the specific details. □

By opting for $q\to 1^{-}$ in Theorem 2, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{H}}_{\sigma }^k(m,\mu )$.

Corollary 3. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{H}}_{\sigma }^k(m,\mu )$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{m(1-\mu )}{3.3^k}}}$$

$$|a_3|\le {\displaystyle \frac{m(1-\mu )}{3.3^k}}$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\mu )(1-\vartheta )}{3.3^k}}\hfill & \mathit{for}\vartheta \le 0,\hfill \\ {\displaystyle \frac{m(1-\mu )}{3.3^k}}\hfill & \mathit{for}0\le \vartheta \le 2,\hfill \\ {\displaystyle \frac{m(1-\mu )(\vartheta -1)}{3.3^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

By opting for $q\to 1^{-}$ and $k=0$ in Theorem 1, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{H}}_{\sigma }(m,\mu )$.

Corollary 4. 

Assume that $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{H}}_{\sigma }(m,\mu )$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{m(1-\mu )}{3}}}$$

$$|a_3|\le {\displaystyle \frac{m(1-\mu )}{3}}$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\mu )(1-\vartheta )}{3}}\hfill & \mathit{for}\vartheta \le 0,\hfill \\ {\displaystyle \frac{m(1-\mu )}{3}}\hfill & \mathit{for}0\le \vartheta \le 2,\hfill \\ {\displaystyle \frac{m(1-\mu )(\vartheta -1)}{3}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

Remark 3. 

1. Corollary 2 provides a verification for the results established by Breaz [19].

2. When we assign $m=2$ in Corollary 2, it provides a verification for the results established by Sivasubramanian [5].

3. Corollary 4 affirms the bounds on $|a_2|$ and improves the bound on $|a_3|$ that was obtained by Li [20], and Corollary 4 provides a verification for the results established by Sharma [21].

4. When we assign $m=2$ in Corollary 4, it affirms the bound $|a_2|$ and improves the bound $|a_3|$ that was obtained by Srivastava [6].

Definition 2. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. Let a function $h\in \mathcal{A}$ be defined by (1). We denote h as belonging to the class ${\mathcal{Q}}_{\sigma }^k(m,\mu ,q)$ if there exist two convex functions, $g\left(\nu \right)$ and $\psi \left(u\right)$, which satisfy the following conditions:

$${\displaystyle \frac{{\mathcal{D}}_q^{k+2}h\left(\nu \right)}{{\mathcal{D}}_q^{k+1}g\left(\nu \right)}}\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\displaystyle \frac{{\mathcal{D}}_q^{k+2}\varphi \left(u\right)}{{\mathcal{D}}_q^{k+1}\psi \left(u\right)}}\in {\mathcal{P}}_m\left(\mu \right).$$

Remark 4. 

1. By opting for $q\to 1^{-}$, the class ${\mathcal{Q}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{Q}}_{\sigma }^{*}(m,\mu )$. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$${\displaystyle \frac{{\left(\nu {\left({\mathcal{D}}^{k}h\right)}^{\prime }\right)}^{\prime }\left(\nu \right)}{{\left({\mathcal{D}}^{k}g\right)}^{\prime }\left(\nu \right)}}\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\displaystyle \frac{{\left(u{\left({\mathcal{D}}^k\varphi \right)}^{\prime }\right)}^{\prime }\left(u\right)}{{\left({\mathcal{D}}^k\psi \right)}^{\prime }\left(u\right)}}\in {\mathcal{P}}_m\left(\mu \right).$$

2. By opting for $q\to 1^{-}$ and $k=0$, the class ${\mathcal{Q}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{Q}}_{\sigma }^{*}\left(\mu \right)$, which was established by Sharma [22] in 2024. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$${\displaystyle \frac{{\left(\nu h^{\prime }\left(\nu \right)\right)}^{\prime }}{g^{\prime }\left(\nu \right)}}\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\displaystyle \frac{{\left(u{\varphi }^{\prime }\left(u\right)\right)}^{\prime }}{{\psi }^{\prime }\left(u\right)}}\in {\mathcal{P}}_m\left(\mu \right).$$

3. By opting for $q\to 1^{-}$, $k=0$ and $m=2$, the class ${\mathcal{Q}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{Q}}_{\sigma }^k(m,\mu )$, which was established by Sharma [22] in 2024. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$$\Re \left({\displaystyle \frac{{\left(\nu h^{\prime }\left(\nu \right)\right)}^{\prime }}{g^{\prime }\left(\nu \right)}}\right)>\mu$$

and

$$\Re \left({\displaystyle \frac{{\left(u{\varphi }^{\prime }\left(u\right)\right)}^{\prime }}{{\psi }^{\prime }\left(u\right)}}\right)>\mu .$$

Theorem 3. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{Q}}_{\sigma }^k(m,\mu ,q)$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{9{\left[3\right]}_q^k}}},$$

(26)

$$|a_3|\le {\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{9{\left[3\right]}_q^k}}$$

(27)

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{(1-\vartheta )[3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)]}{9{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta <0,\hfill \\ {\displaystyle \frac{3{\left[3\right]}_q^k(1-\vartheta )+m(1-\mu )(2{\left[2\right]}_q^k(1-\vartheta )+1)}{9{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}0\le \vartheta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{{\left[3\right]}_q^k+m(1-\vartheta )(2{\left[2\right]}_q^k(1-\vartheta )+1)}{9{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \vartheta <1,}\hfill \\ {\displaystyle \frac{{\left[3\right]}_q^k+m(\vartheta -1)(2{\left[2\right]}_q^k(\vartheta -1)+1)}{9{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}1\le \vartheta \le \frac{4}{3},}\hfill \\ {\displaystyle \frac{3{\left[3\right]}_q^k(\vartheta -1)+m(1-\mu )(2{\left[2\right]}_q^k(\vartheta -1)+1)}{9{\left[3\right]}_q^k}}\hfill & {\displaystyle \mathit{for}\frac{4}{3}<\vartheta <2,}\hfill \\ {\displaystyle \frac{(\vartheta -1)[3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)]}{9{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

(28)

Proof. 

Consider $h\in {\mathcal{Q}}_{\sigma }^k(m,\mu ,q)$. Based on Definition 2, we can define two analytic functions, $t\left(\nu \right)$ and $y\left(u\right)$, such that $t\left(0\right)=y\left(0\right)=1$. These functions belong to the set ${\mathcal{P}}_m\left(\mu \right)$ and fulfill the following conditions:

$${\displaystyle \frac{{\mathcal{D}}_q^{k+2}h\left(\nu \right)}{{\mathcal{D}}_q^{k+1}g\left(\nu \right)}}=t\left(\nu \right)$$

(29)

and

$${\displaystyle \frac{{\mathcal{D}}_q^{k+2}\varphi \left(u\right)}{{\mathcal{D}}_q^{k+1}\psi \left(u\right)}}=y\left(u\right),$$

(30)

where $t\left(\nu \right)$ and $y\left(u\right)$ are given in the forms of (10) and (11). Equations (29) and (30) may be reformulated as follows:

$${\mathcal{D}}_q^{k+2}h\left(\nu \right)={\mathcal{D}}_q^{k+1}g\left(\nu \right)t\left(\nu \right)$$

(31)

and

$${\mathcal{D}}_q^{k+2}\varphi \left(u\right)={\mathcal{D}}_q^{k+1}\psi \left(u\right)y\left(u\right).$$

(32)

Consequently, through a comparison of Equations (10), (11), (31), and (32), we obtain

$$4{\left[2\right]}_q^k{a}_2=2{\left[2\right]}_q^k{d}_2+t_1,$$

(33)

$$9{\left[3\right]}_q^k{a}_3=3{\left[3\right]}_q^k{d}_3+2{\left[2\right]}_q^k{a}_2{t}_1+t_2,$$

(34)

$$-4{\left[2\right]}_q^k{a}_2=-2{\left[2\right]}_q^k{d}_2+y_1$$

(35)

and

$$18{\left[3\right]}_q^k{a}_2^2-9{\left[3\right]}_q^k{a}_3=6{\left[3\right]}_q^k{d}_2^2-3{\left[3\right]}_q^k{d}_3-2{\left[2\right]}_q^k{a}_2{y}_1+y_2.$$

(36)

Therefore, by summing Equations (33) and (35), we obtain $t_1+y_1=0$. Additionally, by adding Equations (34) and (36) and utilizing the relationship $t_1+y_1=0$, we arrive at the following result:

$$18{\left[3\right]}_q^k{a}_2^2=6{\left[3\right]}_q^k{d}_2^2+4{\left[2\right]}_q^k{a}_2{t}_1+t_2+y_2.$$

(37)

Through the application of Lemmas 1 and 2 in Equation (37), we obtain the following outcome:

$$|a_2{|}^2\le {\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{9{\left[3\right]}_q^k}}.$$

(38)

According to Equation (38), the bound of $|a_2|$ is specified in Equation (26). Similarly, through the application of Lemmas 1 and 2 in Equation (34), we obtain the following outcome:

$$|a_3|\le {\displaystyle \frac{3{\left[3\right]}_q^k+m(1-\mu )(2{\left[2\right]}_q^k+1)}{9{\left[3\right]}_q^k}}.$$

(39)

According to Equation (39), the bound of $|a_3|$ is specified in Equation (27). Based on Equations (34) and (37), it follows that for any real number $\vartheta$, we obtain the following result:

$$a_3-\vartheta a_2^2={\displaystyle \frac{6{\left[3\right]}_q^k(d_3-\vartheta d_2^2)+4{\left[2\right]}_q^k{d}_2{t}_1(1-\vartheta )+t_2(2-\vartheta )-\vartheta y_2}{18{\left[3\right]}_q^k}}.$$

(40)

Through the application of Lemmas 1 and 2 in Equation (40), we obtain the following outcome:

$$|a_3-\vartheta a_2^2|\le {\displaystyle \frac{6{\left[3\right]}_q^k|d_3-\vartheta d_2^2{|+4\left[2\right]}_q^{k}m(1-\mu )|1-\vartheta |+m(1-\mu )[|2-\vartheta |+|\vartheta |]}{18{\left[3\right]}_q^k}}.$$

(41)

Employing Lemma 3 within the context of Equation (41) leads us to the bound of $|a_3-\vartheta a_2^2|$, which is specified in Equation (7). This signifies the end of the proof for Theorem 3. □

By opting for $q\to 1^{-}$ in Theorem 3, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{Q}}_{\sigma }^k(m,\mu )$.

Corollary 5. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{Q}}_{\sigma }^k(m,\mu )$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{3.3^k+m(1-\mu )(2.2^k+1)}{9.3^k}}},$$

$$|a_3|\le {\displaystyle \frac{3.3^k+m(1-\mu )(2.2^k+1)}{9.3^k}}$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{(1-\vartheta )[3.3^k+m(1-\mu )(2.2^k+1)]}{9.3^k}}\hfill & \mathit{for}\vartheta <0,\hfill \\ {\displaystyle \frac{3.3^k(1-\vartheta )+m(1-\mu )(2.2^k(1-\vartheta )+1)}{9.3^k}}\hfill & {\displaystyle \mathit{for}0\le \vartheta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{3^k+m(1-\vartheta )(2.2^k(1-\vartheta )+1)}{9.3^k}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \vartheta <1,}\hfill \\ {\displaystyle \frac{3^k+m(\vartheta -1)(2.2^k(\vartheta -1)+1)}{9.3^k}}\hfill & {\displaystyle \mathit{for}1\le \vartheta \le \frac{4}{3},}\hfill \\ {\displaystyle \frac{3.3^k(\vartheta -1)+m(1-\mu )(2.2^k(\vartheta -1)+1)}{9.3^k}}\hfill & {\displaystyle \mathit{for}\frac{4}{3}<\vartheta <2,}\hfill \\ {\displaystyle \frac{(\vartheta -1)[3.3^k+m(1-\mu )(2.2^k+1)]}{9.3^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

By opting for $q\to 1^{-}$ and $k=0$ in Theorem 3, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{Q}}_{\sigma }(m,\mu )$.

Corollary 6. 

Assume that $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{Q}}_{\sigma }(m,\mu )$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{1+m(1-\mu )}{3}}},$$

$$|a_3|\le {\displaystyle \frac{1+m(1-\mu )}{3}}$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{(1-\vartheta )[1+m(1-\mu \left)\right]}{3}}\hfill & \mathit{for}\vartheta <0,\hfill \\ {\displaystyle \frac{3(1-\vartheta )+m(1-\mu )\left(2\right(1-\vartheta )+1)}{9}}\hfill & {\displaystyle \mathit{for}0\le \vartheta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{1+m(1-\vartheta )\left(2\right(1-\vartheta )+1)}{9}}\hfill & {\displaystyle \mathit{for}\frac{2}{3}\le \vartheta <1,}\hfill \\ {\displaystyle \frac{1+m(\vartheta -1)\left(2\right(\vartheta -1)+1)}{9}}\hfill & {\displaystyle \mathit{for}1\le \vartheta \le \frac{4}{3},}\hfill \\ {\displaystyle \frac{3(\vartheta -1)+m(1-\mu )\left(2\right(\vartheta -1)+1)}{9}}\hfill & {\displaystyle \mathit{for}\frac{4}{3}<\vartheta <2,}\hfill \\ {\displaystyle \frac{(\vartheta -1)[1+m(1-\mu \left)\right]}{3}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

The choice of the function $g\left(\nu \right)=\nu$ in Definition 2 leads to a significant reduction of the class ${\mathcal{Q}}_{\sigma }^k(m,\mu ,q)$ to the class ${\mathcal{F}}_{\sigma }^k(m,\mu ,q)$. Functions h within the class ${\mathcal{F}}_{\sigma }^k(m,\mu ,q)$ are those that are part of $\sigma$ and meet the following specified requirements:

$${\mathcal{D}}_q^{k+2}h\left(\nu \right)\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\mathcal{D}}_q^{k+2}\varphi \left(u\right)\in {\mathcal{P}}_m\left(\mu \right).$$

Remark 5. 

1. By opting for $q\to 1^{-}$, the class ${\mathcal{F}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{F}}_{\sigma }^k(m,\mu )$. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$${\left(\nu {\left({\mathcal{D}}^{k}h\right)}^{\prime }\right)}^{\prime }\left(\nu \right)\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\left(u{\left({\mathcal{D}}^k\varphi \right)}^{\prime }\right)}^{\prime }\left(u\right)\in {\mathcal{P}}_m\left(\mu \right).$$

2. By opting for $q\to 1^{-}$ and $k=0$, the class ${\mathcal{F}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{F}}_{\sigma }(m,\mu )$, which was established by Sharma [22] in 2024. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$$h^{\prime }\left(\nu \right)+\nu h^{\prime }\left(\nu \right)\in {\mathcal{P}}_m\left(\mu \right)$$

and

$${\varphi }^{\prime }\left(u\right)+u{\varphi }^{\prime }\left(u\right)\in {\mathcal{P}}_m\left(\mu \right).$$

3. By opting for $q\to 1^{-}$, $k=0$ and $m=2$, the class ${\mathcal{F}}_{\sigma }^k(m,\mu ,q)$ leads to the family ${\mathcal{F}}_{\sigma }\left(\mu \right)$, which was established by Srivastava [23] in 2015. This family consists of the functions $h\in \mathcal{A}$ that fulfill the requirements

$$\Re \left(h^{\prime }\left(\nu \right)+\nu h^{\prime }\left(\nu \right)\right)>\mu$$

and

$$\Re \left({\varphi }^{\prime }\left(u\right)+u{\varphi }^{\prime }\left(u\right)\right)>\mu .$$

The proof of Theorem 4 is straightforward; hence, we chose to omit specific details.

Theorem 4. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{F}}_{\sigma }^k(m,\mu ,q)$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{m(1-\mu )}{9{\left[3\right]}_q^k}}}$$

(42)

$$|a_3|\le {\displaystyle \frac{m(1-\mu )}{9{\left[3\right]}_q^k}}$$

(43)

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\mu )(1-\vartheta )}{9{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta \le 0,\hfill \\ {\displaystyle \frac{m(1-\mu )}{9{\left[3\right]}_q^k}}\hfill & \mathit{for}0\le \vartheta \le 2,\hfill \\ {\displaystyle \frac{m(1-\mu )(\vartheta -1)}{9{\left[3\right]}_q^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

(44)

By opting for $q\to 1^{-}$ in Theorem 4, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{F}}_{\sigma }^k(m,\mu )$.

Corollary 7. 

Assume that $k\in \mathbb{N}\cup \left\{0\right\}$, $0\le \mu <1$ and $2\le m\le 4$. A function $h\in {\mathcal{F}}_{\sigma }^k(m,\mu )$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{m(1-\mu )}{9.3^k}}}$$

$$|a_3|\le {\displaystyle \frac{m(1-\mu )}{9.3^k}}$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\mu )(1-\vartheta )}{9.3^k}}\hfill & \mathit{for}\vartheta \le 0,\hfill \\ {\displaystyle \frac{m(1-\mu )}{9.3^k}}\hfill & \mathit{for}0\le \vartheta \le 2,\hfill \\ {\displaystyle \frac{m(1-\mu )(\vartheta -1)}{9.3^k}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

By opting for $q\to 1^{-}$ and $k=0$ in Theorem 4, we can derive the subsequent corollary that is applicable to functions within the class ${\mathcal{F}}_{\sigma }(m,\mu )$.

Corollary 8. 

Assume that $0\le \mu <1$ and $2\le m\le 4$. If a function $h\in {\mathcal{F}}_{\sigma }(m,\mu )$, then

$$|a_2|\le \sqrt{{\displaystyle \frac{m(1-\mu )}{9}}}$$

$$|a_3|\le {\displaystyle \frac{m(1-\mu )}{9}}$$

and for every $\vartheta \in \mathbb{R}$, we have

$$|a_3-\vartheta a_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{m(1-\mu )(1-\vartheta )}{9}}\hfill & \mathit{for}\vartheta \le 0,\hfill \\ {\displaystyle \frac{m(1-\mu )}{9}}\hfill & \mathit{for}0\le \vartheta \le 2,\hfill \\ {\displaystyle \frac{m(1-\mu )(\vartheta -1)}{9}}\hfill & \mathit{for}\vartheta \ge 2.\hfill \end{array}\right.$$

Remark 6. 

1. Corollary 6 provides a verification for the results established by Sharma [22].

2. When we assign $m=2$ in Corollary 6, it provides a verification for the results established by Sharma [22].

3. Corollary 8 provides a verification for the results established by Sharma [21].

4. When we assign $m=2$ in Corollary 8, it affirms the bounds on $|a_2|$ and improves the bounds on $|a_3|$ that were obtained by Srivastava [23].

3. Concluding Remarks and Observations

In this work, we initially identified the two leading Taylor–Maclaurin coefficients for new subclasses of bi-univalent functions with bounded boundary rotation in the open unit disk $\mathbb{E}$, involving the q-Sălăgean operator. Additionally, we established the notable Fekete–Szegö inequality for these subclasses. We also provided relevant remarks on the main findings, including enhancements to previously established bounds.

Furthermore, the study considered in this article can be extended by taking the q-analogue of a Bessel function, q-analogue of a Mittag–Leffler-type function, a q-exponential function and a q-Ruscheweyh derivative with bounded boundary rotation and bounded radius rotation. However, these interesting details and observations are not addressed. Moreover, the same type of results can be obtained for other interesting special functions found in the literature.