Coefficient Estimates and Symmetry Analysis for Certain Families of Bi-Univalent Functions Defined by the q-Bernoulli Polynomial

Abstract

In the present work, we define certain families, ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ and ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$, of normalized holomorphic and bi-univalent functions associated with Bazilevič functions and $\gimel$-pseudo functions involving the $q$-Bernoulli polynomial, which is defined by the symmetric nature of quantum calculus in the open unit disk $U$. We determine the upper bounds for the initial symmetry Taylor–Maclaurin coefficients and the Fekete–Szegö-type inequalities of functions in the families we have introduced here. In addition, we indicate certain special cases and consequences for our results.

1. Introduction

We denote by $\mathcal{A}$ the collection of functions which are analytic in the open unit disk

$$U=\left\{\mathbb{ᶎ}\in \mathbb{C}:\left|\mathbb{ᶎ}\right|<1\right\}$$

where $\mathbb{C}$ represents the field of complex numbers and has the following normalized form:

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

(1)

Let $S$ be the sub-collection of the set $\mathcal{A}$ consisting of functions which are also univalent in $U$.

A function $f\in S$ is called starlike of order $\rho \left(0\le \rho <1\right),$ if

$$Re\left\{{\displaystyle \frac{\mathbb{ᶎ}{f}^{\prime }\left(\mathbb{ᶎ}\right)}{f\left(\mathbb{ᶎ}\right)}}\right\}>\rho , \left(\mathbb{ᶎ}\in U\right)$$

and a function $f \in S$ is called convex of order $\rho \left(0\le \rho <1\right),$ if

$$Re\left\{{\displaystyle \frac{\mathbb{ᶎ}{f}^{″}\left(\mathbb{ᶎ}\right)}{f^{\prime }\left(\mathbb{ᶎ}\right)}}+1\right\}>\rho , \left(\mathbb{ᶎ}\in U\right).$$

These are standard geometric conditions that guarantee, respectively, starlikeness and convexity; they are expressed via (logarithmic-type) derivatives and measure image-domain geometry.

We define $S^{\ast }(\rho )$ and $C(\rho )$ as the families of functions that are starlike of order ρ and convex of order ρ in $U$, correspondingly.

A function $f\in \mathcal{A}$ is called a Bazilevič function in $U$ if (see [1])

$$Re\left({\displaystyle \frac{{\mathbb{ᶎ}}^{1-{\rm Y}}{f}^{\prime }(\mathbb{ᶎ})}{{\left(f(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}\right)>0 (\mathbb{ᶎ}\in U;{\rm Y}\ge 0).$$

A function $f\in \mathcal{A}$ is designated as a $\gimel$-pseudo-starlike function within $U$ if (see [2])

$$Re\left\{{\displaystyle \frac{\mathbb{ᶎ}{\left(f^{\prime }(\mathbb{ᶎ})\right)}^{\gimel }}{f(\mathbb{ᶎ})}}\right\}>0, \left(\mathbb{ᶎ}\in U;\gimel \ge 1\right).$$

Recently, several authors have introduced and studied different subfamilies associated with Bazilevič and $\gimel$-pseudo functions (see, for example, [3,4,5]).

According to the Koebe one-quarter theorem [6], each function $f \in S$ has an inverse $f^{-1}$ defined by

$$f^{-1}\left(f\left(\mathbb{ᶎ}\right)\right)=\mathbb{ᶎ}, \left(\mathbb{ᶎ}\in U\right)$$

and

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

where

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

(2)

A function $f\in \mathcal{A}$ is termed bi-univalent in $U$ if both f and its inverse $f^{-1}$ are univalent in $U$. The collection of all bi-univalent functions in $U$ is denoted by $\Sigma$. For a concise historical overview and other intriguing instances of functions within the family $\Sigma$, one may refer to the seminal research conducted by Srivastava et al. [7]. A substantial number of sequels to the aforementioned work of Srivastava et al. [7] have introduced and examined numerous studies related to bi-univalent functions across various subfamilies by multiple authors (see, for instance, [8,9,10,11,12,13]). In this direction, more recent contributions have been devoted to the investigation of coefficient bounds for various subclasses of bi-univalent functions. For example, [14] studied families of bi-univalent functions with missing coefficients and obtained several sharp coefficient estimates. Their results further highlight the growing interest in estimating initial coefficients and functionals such as the Fekete–Szegö problem for different subclasses of bi-univalent functions. We will refer to the paragraph that follows the examples of functions from the family $\Sigma$, as presented by Srivastava et al. [7] and shown below in

Table 1. Some functions in the class Σ along with their corresponding inverses.

Functions	Inverse Functions
$f_1\left(\mathbb{ᶎ}\right)={\displaystyle \frac{\mathbb{ᶎ}}{1-\mathbb{ᶎ}}}$	$f_1^{-1}\left(w\right)={\displaystyle \frac{w}{1+w}}$
$f_2\left(\mathbb{ᶎ}\right)=-\mathit{log}\left(1-\mathbb{ᶎ}\right)$	$f_2^{-1}\left(w\right)=1-e^{-w}$
$f_3\left(\mathbb{ᶎ}\right)={\displaystyle \frac{1}{2}}log\left({\displaystyle \frac{1+\mathbb{ᶎ}}{1-\mathbb{ᶎ}}}\right)$	$f_3^{-1}\left(w\right)={\displaystyle \frac{e^{2w}-1}{e^{2w}+1}}$

.

The family $\Sigma$ is non-empty. Nonetheless, the Koebe function does not belong to $\Sigma$. The issue of determining the universal boundaries for the Taylor–Maclaurin coefficients is defined here as

$$\left|a_n\right| \left(n\in \mathbb{N};n\ge 3\right)$$

The issue of functions $f \in \Sigma$ remains inadequately resolved for numerous subfamilies of the bi-univalent function family $\Sigma$.

The Fekete–Szegö functional $\left|a_3-\mu a_2^2\right|$ for $f\in S$ is renowned for its extensive history in the domain of Geometric Function Theory. The foundation lies in the refutation by Fekete and Szegö [15] of the Littlewood–Paley conjecture, which posited that the coefficients of odd univalent functions are constrained to unity. The functional has garnered significant interest, especially in the examination of several subfamilies within the family of univalent functions. This subject has garnered significant attention among scholars in the Geometric Function Theory of Complex Analysis.

To reiterate the principle of subordination among holomorphic functions, assume the functions both $f$ and $g$ to be holomorphic in $U$. The function $f$ is considered subordinate to $g$ once there exists a Schwarz function $w$, which is analytic in $U$, satisfying $w\left(0\right)=0$ as well as $\left|w(\mathbb{ᶎ})\right|<1$ for all $\mathbb{ᶎ}$ in $U$, such that

$$f\left(\mathbb{ᶎ}\right)=g\left(w\left(\mathbb{ᶎ}\right)\right).$$

This subordination is denoted by

$$f\prec g or f\left(\mathbb{ᶎ}\right)\prec g\left(\mathbb{ᶎ}\right) (\mathbb{ᶎ}\in U)$$

It is recognized by many that if the function $g$ is univalent within $U$, then (refer to [16])

$$f\prec g \left(\mathbb{ᶎ}\in U\right) ⟺ f\left(0\right)=g\left(0\right) \mathrm{a}\mathrm{n}\mathrm{d} f\left(U\right)\subseteq g\left(U\right).$$

For $0<q<1$, the q-factorial denoted by ${\left[n\right]}_q!$ is defined by (see [17])

$${\left[n\right]}_q!=\left\{\begin{array}{c}{\left[n\right]}_q{\left[n-1\right]}_q\dots {\left[2\right]}_q{\left[1\right]}_q, if n=1, 2, 3, \dots ,\\ 1, if n=0, \end{array}\right.$$

where ${\left[n\right]}_q$, called the $q$-analog of $n \in \mathbb{N}$, is given by

$${\left[n\right]}_q={\displaystyle \frac{1-q^n}{1-q}} for n\in \mathbb{N},$$

Jackson [17,18] introduced the q-derivative operator ${\mathfrak{D}}_q$ of a function $f$ as follows:

$${\mathfrak{D}}_{q}f\left(\mathbb{ᶎ}\right)={\displaystyle \frac{f\left(\mathbb{ᶎ}\right)-f\left(q\mathbb{ᶎ}\right)}{{\left(1-q\right)}^{\mathbb{ᶎ}}}} \left(0<q<1;\mathbb{ᶎ}\ne 0\right).$$

It is clear that

$$\underset{q\to 1-}{\mathit{lim}}{\mathfrak{D}}_{q}f\left(\mathbb{ᶎ}\right)=f^{\prime }\left(\mathbb{ᶎ}\right) \mathrm{a}\mathrm{n}\mathrm{d} {\mathfrak{D}}_{q}f\left(0\right)=f^{\prime }\left(0\right).$$

For additional conceptual information regarding the $q$-derivative operator ${\mathfrak{D}}_q$, refer to [19,20,21].

Using a function $f\in \mathcal{A}$ described by (1), we infer that

$${\mathfrak{D}}_{q}f\left(\mathbb{ᶎ}\right)=1+\sum _{n=2}^{\infty }{\left[n\right]}_q{a}_n{\mathbb{ᶎ}}^{n-1},$$

As $q\to 1-$, then we have ${\left[n\right]}_q\to n$ and ${\left[0\right]}_q$ = 0.

The q-exponential function $e_q$ is defined by its power series expansion (see [22])

$$e_q\left(\mathbb{ᶎ}\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\mathbb{ᶎ}}^n}{{\left[n\right]}_q!}}, \left(\mathbb{ᶎ}\in U\right).$$

We note that

$$e\left(\mathbb{ᶎ}\right)=\underset{q\to 1-}{lim}{e}_q\left(\mathbb{ᶎ}\right)=\sum _{n=0}^{\infty }{\displaystyle \frac{{\mathbb{ᶎ}}^n}{\left[n\right]!}}.$$

The $q$-exponential function $e_q$ is a unique function that satisfies the condition

$${\displaystyle \frac{{\mathfrak{D}}_{q}e\left(\mathbb{ᶎ}\right)}{{\mathfrak{D}}_q\mathbb{ᶎ}}}=\sum _{n=0}^{\infty }{\displaystyle \frac{{\mathfrak{D}}_q{\mathbb{ᶎ}}^n}{{\left[n\right]}_q!}}=\sum _{n=1}^{\infty }{\displaystyle \frac{{{\left[n\right]}_q\mathbb{ᶎ}}^{n-1}}{{\left[n\right]}_q!}}=\sum _{n=1}^{\infty }{\displaystyle \frac{{\mathbb{ᶎ}}^{n-1}}{{\left[n-1\right]}_q!}}=\sum _{n=0}^{\infty }{\displaystyle \frac{{\mathbb{ᶎ}}^n}{{\left[n\right]}_q!}}=e_q\left(\mathbb{ᶎ}\right), \left(\mathbb{ᶎ}\in U\right).$$

In recent years, numerous authors have examined various applications of q-calculus in relation to different families of analytic as well as univalent (or multivalent) functions (see, for instance, [23,24]). In his recently released survey and explanatory review paper, Srivastava [12] examined the mathematical applications of $q$-calculus, fractional $q$-calculus, and fractional $q$-derivative operators within the realm of the Geometric Function Theory of Complex Analysis. Srivastava [25] elucidated the not-yet-commonly comprehended notion that the ($p,q$)-variation in traditional $q$-calculus is a very trivial and unimportant modification, with the additional parameter $p$ being redundant or unneeded (see [25] (p. 340) for details).

According to the symmetric nature of quantum calculus, the $q$-Bernoulli polynomials ${\mathfrak{B}}_{q,n}(x)$ in the Geometric Function Theory of Complex Analysis are given by the following linear homogeneous recurrence relation (see, for instance, [26,27]):

$${\mathfrak{B}}_{q,n}\left(x\right)=q^n\left(x-{\displaystyle \frac{1}{{q\left[2\right]}_q}}\right){\mathfrak{B}}_{q,n-1}\left(x\right)-{\displaystyle \frac{1}{{\left[n\right]}_q}} \sum _{j=0}^{n-2}{\left({\displaystyle \genfrac{}{}{0pt}{}{n}{j}}\right)}_q{q}^{j-1}{b}_{n-j,q}{\mathfrak{B}}_{q,n}\left(x\right),$$

(3)

with

$${\mathfrak{B}}_{q,0}\left(x\right)=1, {\mathfrak{B}}_{q,1}\left(x\right)={\displaystyle \frac{{\left[2\right]}_{q}x-q}{{\left[2\right]}_q}} , \mathrm{a}\mathrm{n}\mathrm{d} {\mathfrak{B}}_{q,2}\left(x\right)=x\left(x-1\right)+{\displaystyle \frac{q}{{\left[2\right]}_{q }{\left[3\right]}_q}}.$$

In order to illustrate the behavior of the q-Bernoulli polynomials, we refer to Figure 1 and Figure 2. Figure 1 provides two separate visualizations: the left panel displays the polynomial ${\mathfrak{B}}_{\mathrm{3,0.5}}\left(x\right)$ as a function of $x$ for the fixed value $q=0.5$, while the right panel shows ${\mathfrak{B}}_{q,3}\left(1\right)$ as a function of q for the fixed value $x=1$. To further explore the variation of the q-Bernoulli polynomials with respect to both variables simultaneously, Figure 2 depicts their three-dimensional behavior over $x$ and $q$, offering a comprehensive graphical representation of their joint dependence.

The generating function of the $q$-Bernoulli polynomials ${\mathfrak{B}}_{q,n}(x)$ is given as follows (see [26]):

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

(4)

The families of orthogonal polynomials, other special functions, and specific polynomials, along with their expansions and generalizations, hold potential utility across various scientific fields, particularly within the mathematical, statistical, and physical sciences. The relationship between bi-univalent functions and orthogonal polynomials has recently come under the scrutiny of various authors (see, for example, [28,29,30,31,32]).

2. Main Results

Using the $q$-Bernoulli polynomials, we now define the following families, ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ and ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$, of holomorphic bi-starlike and bi-convex functions.

Definition 1.

A function $f\in \Sigma$ is classified inside the family  ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ if it meets the subsequent subordination criteria:

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{1-{\rm Y}}{f}^{\prime }(\mathbb{ᶎ})}{{\left(f(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{\mathbb{ᶎ}{\left(f^{\prime }(\mathbb{ᶎ})\right)}^{\gimel }}{f(\mathbb{ᶎ})}} \prec {\mathfrak{B}}_{q,n}\left(x,\mathbb{ᶎ}\right)$$

and

$$\left(1-\mu \right){\displaystyle \frac{w^{1-{\rm Y}}{g}^{\prime }(w)}{{\left(g(w)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{w{\left(g^{\prime }(w)\right)}^{\gimel }}{g(w)}}\prec {\mathfrak{B}}_{q,n}\left(x,w\right),$$

where $0\le \mu \le 1, {\rm Y}\ge 0, \gimel \ge 1, \mathbb{ᶎ}, w \in U, x \in [-\pi , \pi ]$ and the function  $g=f^{-1}$ is given by (2).

Remark 1.

The family${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ is non-empty. For example, the identity function  $f\left(\mathbb{ᶎ}\right)=\mathbb{ᶎ}$ (with inverse$g\left(w\right)=w$) satisfies both subordinations, since each condition reduces to the constant function (1), which belongs to the range of  ${\mathfrak{B}}_{q,n}\left(x,0\right)$ for suitable parameters.

Remark 2.

If we take  $\mu ={\rm Y}=0$ in Definition 1, the family  ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ reduces to the family  $S_{\Sigma }^{\ast }\left(q; x\right),$ which was studied recently by Wanas and Khachi (see [5]).

Definition 2.

A function  $f\in \Sigma$ is said to be in the family  ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ if it fulfills the following subordination conditions:

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{2-{\rm Y}}{f}^{″}(\mathbb{ᶎ})}{{\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{\prime }\right)}^{\gimel }}{f^{\prime }(\mathbb{ᶎ})}}\prec {\mathfrak{B}}_{q,n}\left(x,\mathbb{ᶎ}\right)$$

and

$$\left(1-\mu \right){\displaystyle \frac{w^{2-{\rm Y}}{g}^{″}\left(w\right)}{{\left(w{g}^{\prime }\left(w\right)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }\right)}^{\gimel }}{g^{\prime }\left(w\right)}}\prec {\mathfrak{B}}_{q,n}\left(x,w\right),$$

where $0\le \mu \le 1, {\rm Y}\ge 0, \gimel \ge 1, \mathbb{ᶎ}, w\in U, x \in [-\pi , \pi ]$ and the function  $g=f^{-1}$ is given by (2).

For brevity, denote $\alpha ={\rm Y}-1$. Write $f\left(\mathbb{ᶎ}\right)={\mathbb{ᶎ}+a}_2{\mathbb{ᶎ}}^2+a_3{\mathbb{ᶎ}}^3+a_4{\mathbb{ᶎ}}^4+\cdots$. A compact computation (factor $f(\mathbb{ᶎ})=\mathbb{ᶎ}({1+a}_2\mathbb{ᶎ}+a_3{\mathbb{ᶎ}}^2+\cdots )$) gives the following series for the defining quantity

$$E\left(\mathbb{ᶎ}\right):=\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{1-{\rm Y}}{f}^{\prime }\left(\mathbb{ᶎ}\right)}{{\left(f\left(\mathbb{ᶎ}\right)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{\mathbb{ᶎ}{\left(f^{\prime }\left(\mathbb{ᶎ}\right)\right)}^{\gimel }}{f\left(\mathbb{ᶎ}\right)}}={1+c}_1\mathbb{ᶎ}+c_2{\mathbb{ᶎ}}^2+O\left({\mathbb{ᶎ}}^3\right),$$

where the coefficients $c_1,c_2$ are

$$c_1=a_2\left[\left(1-\mu \right)\left(2+\alpha \right)+\mu (\gimel -1)\right]$$

and

$$c_2=a_3\left[\left(1-\mu \right)\left(3+\alpha \right)+\mu \left(3\gimel -1\right)\right]+a_2^2\left[\left(1-\mu \right){\displaystyle \frac{\alpha \left(3+\alpha \right)}{2}}+\mu \left(2{\gimel }^2-4\gimel +1\right)\right].$$

To provide concrete illustrations of the families ${\mathbb{M}}_{\Sigma } \mathrm{and} {\mathbb{N}}_{\Sigma }$, some explicit examples are presented in

Table 2. Explicit example functions from the families ${\mathbb{M}}_{\Sigma }$ and ${\mathbb{N}}_{\Sigma }$.

Family	$\mathbf{Parameters} \left(\mathsf{\mu },\mathsf{{\rm Y}},\mathbf{\gimel },\mathbf{q};\mathbf{x}\right)$	$\mathbf{Function} \mathit{f}\left(\mathit{ᶎ}\right)$$(\mathbf{up} \mathbf{to} {\mathbb{ᶎ}}^3)$
${\mathbb{M}}_{\Sigma }$	$(1, 0.5, 1, 1, 0.5)$	$f\left(\mathbb{ᶎ}\right)=\mathbb{ᶎ}+0.4444{\mathbb{ᶎ}}^2+0.11157{\mathbb{ᶎ}}^3$
${\mathbb{N}}_{\Sigma }$	$(0.7, 0.8 , 1.2, 0.5 , 0.8)$	$f\left(\mathbb{ᶎ}\right)=\mathbb{ᶎ}+0.448{\mathbb{ᶎ}}^2+0.052{\mathbb{ᶎ}}^3$

. The listed functions are expanded up to the term ${\mathbb{ᶎ}}^3$ for clarity.

Remark 3.

If we take  $\mu ={\rm Y}=0$ in Definition 2, the family  ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ reduces to the family  $C_{\Sigma }\left(q;x\right),$ which was introduced recently by Wanas and Khachi (see [5]).

The subfamilies extend known families, link with Bernoulli polynomials, and contribute to coefficient problems (e.g., Fekete–Szegö functional).

Theorem 1.

Let  $f\in \mathcal{A}$ be in the family  ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$. Then

$$\left|a_2\right|\le min\left\{\left|{\displaystyle \frac{{\left[2\right]}_{q}x-q}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}}\right|,\right.$$

$$\left.{\displaystyle \frac{2\left|{\left[2\right]}_{q}x-q\right|\sqrt{\left|{\left[2\right]}_{q}x-q\right|}}{{\left[2\right]}_q\sqrt{\left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right|}}}\right\}$$

and

$$\begin{gathered} \left|a_3\right|\le min\left\{{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{2}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]}}\left(\left|{\left[2\right]}_{q}x-q\right| \\ \hspace{1em}\hspace{1em}\hspace{1em} +\left|x\left(x-1\right)\right|+{\displaystyle \frac{q}{{\left[2\right]}_q {\left[3\right]}_q}}\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{q}{{\left[2\right]}_q^2 {\left[3\right]}_q}},{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{{\left({\left[2\right]}_{q}x-q\right)}^2}{{\left[2\right]}_q^2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2}} \right\}. \end{gathered}$$

Proof. 

Assume $f$ belongs to ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$. There exist two holomorphic functions, $u$ and $v:U\to U$, defined by

$$u\left(x\right)=u_1\mathbb{ᶎ}+u_2{\mathbb{ᶎ}}^2+u_3{\mathbb{ᶎ}}^3\dots \left(\mathbb{ᶎ}\in U\right)$$

(5)

and

$$v\left(w\right)=v_{1}w+v_2{w}^2+v_3{w}^3\dots \left(\mathbb{ᶎ}\in U\right) ,$$

(6)

with

$$u\left(0\right)=v\left(0\right)=0 \mathrm{a}\mathrm{n}\mathrm{d} max\left\{\left|u\left(\mathbb{ᶎ}\right)\right|,\left|v(w)\right|\right\}<1 \left(\mathbb{ᶎ},w\in U\right),$$

such that

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{1-{\rm Y}}{f}^{\prime }(\mathbb{ᶎ})}{{\left(f(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{\mathbb{ᶎ}{\left(f^{\prime }(\mathbb{ᶎ})\right)}^{\gimel }}{f(\mathbb{ᶎ})}}={\mathfrak{B}}_q\left(x,u\left(\mathbb{ᶎ}\right)\right)$$

and

$$\left(1-\mu \right){\displaystyle \frac{w^{1-{\rm Y}}{g}^{\prime }(w)}{{\left(g(w)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{w{\left(g^{\prime }(w)\right)}^{\gimel }}{g(w)}}={\mathfrak{B}}_q\left(x,v\left(w\right)\right),$$

or equivalently

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{1-{\rm Y}}{f}^{\prime }(\mathbb{ᶎ})}{{\left(f(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{\mathbb{ᶎ}{\left(f^{\prime }(\mathbb{ᶎ})\right)}^{\gimel }}{f(\mathbb{ᶎ})}}=1+{\mathfrak{B}}_{q,1}\left(x\right)u\left(\mathbb{ᶎ}\right)+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)u^2\left(\mathbb{ᶎ}\right)+\cdots$$

(7)

and

$$\left(1-\mu \right){\displaystyle \frac{w^{1-{\rm Y}}{g}^{\prime }(w)}{{\left(g(w)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{w{\left(g^{\prime }(w)\right)}^{\gimel }}{g(w)}}=1+{\mathfrak{B}}_{q,1}\left(x\right)v\left(w\right)+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)v^2\left(w\right)+\cdots$$

(8)

Combining (5)–(8), we find that

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{1-{\rm Y}}{f}^{\prime }(\mathbb{ᶎ})}{{\left(f(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{\mathbb{ᶎ}{\left(f^{\prime }(\mathbb{ᶎ})\right)}^{\gimel }}{f(\mathbb{ᶎ})}}=1+{\mathfrak{B}}_{q,1}\left(x\right)u_1\mathbb{ᶎ}+\left[{\mathfrak{B}}_{q,1}\left(x\right)u_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)u_1^2\right]{\mathbb{ᶎ}}^2+\cdots$$

(9)

and

$$\left(1-\mu \right){\displaystyle \frac{w^{1-{\rm Y}}{g}^{\prime }(w)}{{\left(g(w)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{w{\left(g^{\prime }(w)\right)}^{\gimel }}{g(w)}}=1{+\mathfrak{B}}_{q,1}\left(x\right)v_{1}w+\left[{\mathfrak{B}}_{q,1}\left(x\right)v_2{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)v_1^2\right]w^2+\cdots .$$

(10)

It is well known that if

$$max \left\{\left|u\left(\mathbb{ᶎ}\right)\right|,\left|v\left(w\right)\right|\right\}<1 ( \mathbb{ᶎ},w\in U),$$

then

$$\left|u_j\right|\le 1 \mathrm{a}\mathrm{n}\mathrm{d} \left|v_j\right|\le 1 \left(\forall j\in \mathbb{N}\right).$$

(11)

By comparing the respective coefficients in (9) as well as (10), then following some simplification, we obtain

$$\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]a_2={\mathfrak{B}}_{q,1}\left(x\right)u_1,$$

(12)

$$\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]a_3+\left[{\displaystyle \frac{1}{2}}\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}-1\right)+\mu (2\gimel \left(\gimel -2\right)+1)\right]a_2^2$$

$$={\mathfrak{B}}_{q,1}\left(x\right)u_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)u_1^2,$$

(13)

$${-\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]a}_2={\mathfrak{B}}_{q,1}\left(x\right)v_1$$

(14)

and

$$\begin{gathered} \left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]({2a}_2^2-a_3)+\left[{\displaystyle \frac{1}{2}}\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}-1\right)+\mu (2\gimel \left(\gimel -2\right)+1)\right]a_2^2 \\ ={\mathfrak{B}}_{q,1}\left(x\right)v_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)v_1^2. \end{gathered}$$

(15)

It follows from (12) and (14) that

$${-u}_1=v_1$$

(16)

and

$${{2\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^{2}a}_2^2={\mathfrak{B}}_{q,1}^2\left(x\right)\left(u_1^2+v_1^2\right).$$

(17)

If we add (13) to (15), we find that

$${\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]a}_2^2={\mathfrak{B}}_{q,1}\left(x\right)\left(u_2+v_2\right)+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)\left(u_1^2+v_1^2\right).$$

(18)

By putting the value of $u_1^2+v_1^2$ from (17) through the right-hand side of (18), we conclude that

$$a_2^2={\displaystyle \frac{{\mathfrak{B}}_{q,1}^3\left(x\right) \left(u_2+v_2\right)}{{\mathfrak{B}}_{q,1}^2\left(x\right)\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-\frac{2}{{\left[2\right]}_q}{\mathfrak{B}}_{q,2}\left(x\right){\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2}}.$$

(19)

Subsequent calculations utilizing (3), (11), (17), and (19) provide

$$\left|a_2\right|\le \left|{\displaystyle \frac{{\left[2\right]}_{q}x-q}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}}\right|,$$

$$\left|a_2\right|\le {\displaystyle \frac{2\left|{\left[2\right]}_{q}x-q\right|\sqrt{\left|{\left[2\right]}_{q}x-q\right|}}{{\left[2\right]}_q\sqrt{\left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right|}}}.$$

Subsequently, by subtracting (15) from (13), it becomes evident that

$${2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]\left(a_3-a_2^2\right)=\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)$$

$$+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)\left(u_1^2-v_1^2\right).$$

(20)

In view of (16) and substituting the value of $a_2^2$ from (17) into (20), we find that

$$a_3={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}+{\displaystyle \frac{{\mathfrak{B}}_{q,1}^2\left(x\right)\left(u_1^2+v_1^2\right)}{{2\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2}}.$$

Thus, by applying (3), we obtain

$$\left|a_3\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}+{\displaystyle \frac{{\left({\left[2\right]}_{q}x-q\right)}^2}{{\left[2\right]}_q^2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2}}.$$

In addition, substituting the value of $a_2^2$ from (18) into (20), we deduce that

$$a_3={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}+{\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2+v_2\right)}{\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]}}$$

$$+{\displaystyle \frac{{\mathfrak{B}}_{q,2}\left(x\right)\left(u_1^2+v_1^2\right)}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]}}$$

and we have

$$\left|a_3\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}$$

$$+{\displaystyle \frac{2}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]}}\left(\left|{\left[2\right]}_{q}x-q\right|+\left|x\left(x-1\right)\right|+{\displaystyle \frac{q}{{\left[2\right]}_q {\left[3\right]}_q}}\right).$$

This completes the proof of Theorem 1. $\square$

When $\mu ={\rm Y} =0$, Theorem 1 is reduced to the corresponding results of Wanas and Khachi (see [5]).

Corollary 1.

[33] If $f$ given by  $(1)$ is in the family  $S_{\Sigma }^{\ast }\left(q; x\right)$, then

$$\left|a_2\right|\le min\left\{\left|{\displaystyle \frac{{\left[2\right]}_{q}x-q}{{\left[2\right]}_q}}\right| ,{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|\sqrt{\left|{\left[2\right]}_{q}x-q\right|}}{{\left[2\right]}_q\sqrt{\left|\left({\left[2\right]}_q-1\right)x^2+\left(1-2q\right)x+\frac{q\left(q{\left[3\right]}_q-1\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right|}}}\right\}$$

and

$$\left|a_3\right|\le min\left\{{\displaystyle \frac{3\left|{\left[2\right]}_{q}x-q\right|}{2{\left[2\right]}_q}}+{\displaystyle \frac{\left|x\left(x-1\right)\right|}{{\left[2\right]}_q}}+{\displaystyle \frac{q}{{\left[2\right]}_q^2 {\left[3\right]}_q}},{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{2{\left[2\right]}_q}}+{\displaystyle \frac{{\left({\left[2\right]}_{q}x-q\right)}^2}{{\left[2\right]}_q^2}}\right\}.$$

Theorem 2.

Let  $f \in \mathcal{A}$ be in the family  ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$. Then

$$\left|a_2\right|\le min\left\{{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{2\left(2\mu \left(\gimel -1\right)+1\right){\left[2\right]}_q}},\right.$$

$$\left.{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right| \sqrt{\left|{\left[2\right]}_{q}x-q\right|}}{{\left[2\right]}_{q }\sqrt{\left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right|}}}\right\}$$

and

$$\begin{gathered} \begin{array}{c}\left|a_3\right|\le min\left\{{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{3{\left[2\right]}_q}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{1}{\left[2+4{\rm Y}+9\mu \left(\gimel -1\right)+8\gimel \mu \left(\gimel -2\right)+4\mu \left(2-{\rm Y}\right)\right]{\left[2\right]}_q}}\left[\left|{\left[2\right]}_{q}x \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-q\right|+\left|x\left(x-1\right)\right|+{\displaystyle \frac{q}{{\left[2\right]}_q {\left[3\right]}_q}}\right],{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{3\left(3\mu \left(\gimel -1\right)+2\right){\left[2\right]}_q}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\displaystyle \frac{{\left(\left|{\left[2\right]}_{q}x-q\right|\right)}^2}{4{\left(2\mu \left(\gimel -1\right)+1\right)}^2{\left[2\right]}_q^2}}\right\}.\end{array} \end{gathered}$$

Proof. 

Suppose that $f\in {\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$. Then there are two holomorphic functions $u,v:U\to U$ such that

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{2-{\rm Y}}{f}^{″}(\mathbb{ᶎ})}{{\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{\prime }\right)}^{\gimel }}{f^{\prime }(\mathbb{ᶎ})}}={\mathfrak{B}}_q\left(x,u\left(\mathbb{ᶎ}\right)\right)$$

and

$$\left(1-\mu \right){\displaystyle \frac{w^{2-{\rm Y}}{g}^{″}\left(w\right)}{{\left(w{g}^{\prime }\left(w\right)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }\right)}^{\gimel }}{g^{\prime }\left(w\right)}}={\mathfrak{B}}_q\left(x,v\left(w\right)\right),$$

where $u$ and $v$ have the forms (5) and (6). We have

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{2-{\rm Y}}{f}^{\prime \prime }(\mathbb{ᶎ})}{{\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{\prime }\right)}^{\gimel }}{f^{\prime }(\mathbb{ᶎ})}}=1+{\mathfrak{B}}_{q,1}\left(x\right)u\left(\mathbb{ᶎ}\right)+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)u^2\left(\mathbb{ᶎ}\right)+\cdots$$

(21)

and

$$\left(1-\mu \right){\displaystyle \frac{w^{2-{\rm Y}}{g}^{\prime \prime }\left(w\right)}{{\left(w{g}^{\prime }\left(w\right)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }\right)}^{\gimel }}{g^{\prime }\left(w\right)}}=1+{\mathfrak{B}}_{q,1}\left(x\right)v\left(w\right)+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)v^2\left(w\right)+\cdots$$

(22)

From (21) and (22), we deduce that

$$\left(1-\mu \right){\displaystyle \frac{{\mathbb{ᶎ}}^{2-{\rm Y}}{f}^{″}(\mathbb{ᶎ})}{{\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(\mathbb{ᶎ}{f}^{\prime }(\mathbb{ᶎ})\right)}^{\prime }\right)}^{\gimel }}{f^{\prime }(\mathbb{ᶎ})}}=1+{\mathfrak{B}}_{q,1}\left(x\right)u_1\mathbb{ᶎ}+\left[{\mathfrak{B}}_{q,1}\left(x\right)u_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)u_1^2\right]{\mathbb{ᶎ}}^2+\cdots$$

(23)

and

$$\left(1-\mu \right){\displaystyle \frac{w^{2-{\rm Y}}{g}^{″}\left(w\right)}{{\left(w{g}^{\prime }\left(w\right)\right)}^{1-{\rm Y}}}}+\mu {\displaystyle \frac{{\left({\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }\right)}^{\gimel }}{g^{\prime }\left(w\right)}}=1{+\mathfrak{B}}_{q,1}\left(x\right)v_{1}w+\left[{\mathfrak{B}}_{q,1}\left(x\right)v_2{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)v_1^2\right]w^2+\cdots$$

(24)

Now, by comparing the corresponding coefficients in (23) and (24), and after some simplification, we have

$$2\left(2\mu \left(\gimel -1\right)+1\right)a_2={\mathfrak{B}}_{q,1}\left(x\right)u_1,$$

(25)

$$3\left(3\mu \left(\gimel -1\right)+2\right)a_3+4\left[2\gimel \mu (\gimel -2)+\mu (2-{\rm Y})+{\rm Y}-1\right]a_2^2$$

$$={\mathfrak{B}}_{q,1}\left(x\right)u_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)u_1^2,$$

(26)

$${-2\left(2\mu \left(\gimel -1\right)+1\right)a}_2={\mathfrak{B}}_{q,1}\left(x\right)v_1$$

(27)

and

$$3\left(3\mu \left(\gimel -1\right)+2\right){(2a_2^2-a}_3)+4\left[2\gimel \mu (\gimel -2)+\mu (2-{\rm Y})+{\rm Y}-1\right]a_3$$

$$={\mathfrak{B}}_{q,1}\left(x\right)v_2+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)v_1^2.$$

(28)

It follows from (25) and (27) that

$${-u}_1=v_1$$

(29)

and

$$8{\left(2\mu \left(\gimel -1\right)+1\right)}^2{a}_2^2={\mathfrak{B}}_{q,1}^2\left(x\right)\left(u_1^2+v_1^2\right) .$$

(30)

If we add (26) to (28), we find that

$$2\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]a_2^2={\mathfrak{B}}_{q,1}\left(x\right)\left(u_2+v_2\right)+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)\left(u_1^2+v_1^2\right).$$

(31)

Upon putting the value of $u_1^2+v_1^2$ given (30) through the right-hand side of (31), we derive that

$$a_2^2={\displaystyle \frac{{\mathfrak{B}}_{q,1}^3\left(x\right) \left(u_2+v_2\right)}{2\left[\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]{\mathfrak{B}}_{q,1}^2\left(x\right)-\frac{4{\left(2\mu \left(\gimel -1\right)+1\right)}^2}{{\left[2\right]}_q}{\mathfrak{B}}_{q,2}\left(x\right)\right]}}.$$

(32)

Subsequent calculations utilizing (3), (11), (30), and (32) provide

$$\left|a_2\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{2\left(2\mu \left(\gimel -1\right)+1\right){\left[2\right]}_q}},$$

$$\left|a_2\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right| \sqrt{\left|{\left[2\right]}_{q}x-q\right|}}{{\left[2\right]}_{q }\sqrt{\left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right|}}} .$$

Subsequently, by subtracting (28) from (26), it becomes evident that

$$6\left(3\mu \left(\gimel -1\right)+2\right)\left(a_3-a_2^2\right)={\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)+{\displaystyle \frac{1}{{\left[2\right]}_q}}{\mathfrak{B}}_{q,2}\left(x\right)\left(u_1^2-v_1^2\right).$$

(33)

Considering (29) and replacing the value of $a_2^2$ to (30) through (33), we ascertain that

$$a_3={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{6\left(3\mu \left(\gimel -1\right)+2\right)}}+{\displaystyle \frac{{\mathfrak{B}}_{q,1}^2\left(x\right)\left(u_1^2+v_1^2\right)}{8{\left(2\mu \left(\gimel -1\right)+1\right)}^2}}.$$

Thus, by applying (3), we obtain

$$\left|a_3\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{3\left(3\mu \left(\gimel -1\right)+2\right){\left[2\right]}_q}}+{\displaystyle \frac{{\left(\left|{\left[2\right]}_{q}x-q\right|\right)}^2}{4{\left(2\mu \left(\gimel -1\right)+1\right)}^2{\left[2\right]}_q^2}}.$$

In addition, by substituting the value of $a_2^2$ from (31) into (33), we deduce that

$$a_3={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{6\left(3\mu \left(\gimel -1\right)+2\right)}}+{\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2+v_2\right)}{2\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]}}+{\displaystyle \frac{{\mathfrak{B}}_{q,2}\left(x\right)\left(u_1^2+v_1^2\right)}{2\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]{\left[2\right]}_q}},$$

and we have

$$\left|a_2\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right| \sqrt{\left|{\left[2\right]}_{q}x-q\right|}}{{\left[2\right]}_{q }\sqrt{\left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right|}}} .$$

$$\left|a_3\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{3{\left[2\right]}_q}}+{\displaystyle \frac{1}{\left[2+4{\rm Y}+9\mu \left(\gimel -1\right)+8\gimel \mu \left(\gimel -2\right)+4\mu \left(2-{\rm Y}\right)\right]{\left[2\right]}_q}}\left[\left|{\left[2\right]}_{q}x-q\right|+\left|x\left(x-1\right)\right|+{\displaystyle \frac{q}{{\left[2\right]}_q {\left[3\right]}_q}}\right].$$

This completes the proof of Theorem 2. $\square$

When $\mu ={\rm Y} =0$, Theorem 2 is reduced to the corresponding results of Wanas and Khachi (see [34]).

Corollary 2.

[35] If $f$ given by  $(1)$ is in the family  $C_{\Sigma }\left(q;x\right)$ , then

$$\left|a_2\right|\le min\left\{{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{2{\left[2\right]}_q}} ,{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|\sqrt{\left|{\left[2\right]}_{q}x-q\right|}}{{\left[2\right]}_q\sqrt{2\left|\left({\left[2\right]}_q-2\right)x^2+2\left(1-q\right)x+\frac{q\left(q{\left[3\right]}_q-2\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right|}}}\right\}$$

and

$$\left|a_3\right|\le min\left\{{\displaystyle \frac{2\left|{\left[2\right]}_{q}x-q\right|}{3{\left[2\right]}_q}}+{\displaystyle \frac{\left|x\left(x-1\right)\right|}{2{\left[2\right]}_q}}+{\displaystyle \frac{q}{2{\left[2\right]}_q^2 {\left[3\right]}_q}},{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{6{\left[2\right]}_q}}+{\displaystyle \frac{{\left({\left[2\right]}_{q}x-q\right)}^2}{4{\left[2\right]}_q^2}}\right\}.$$

In the subsequent theorems, we introduce the Fekete–Szegö-type inequalities for the families ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ and ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$.

Theorem 3.

For  $\mu \in \mathbb{R}$ , let  $f\in \mathcal{A}$ belong to the family  ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ . Subsequently

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}} ; \\ \left|\phi -1\right|\le {\displaystyle \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right| }{2{\left({\left[2\right]}_{q}x-q\right)}^2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}},\\ \frac{2{\left|{\left[2\right]}_{q}x-q\right|}^3\left|\mu -1\right|}{{\left[2\right]}_{q }^3 \left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right|};\\ \left|\phi -1\right|\ge \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right| }{2{\left({\left[2\right]}_{q}x-q\right)}^2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}.\end{array}\right.$$

Proof. 

It is derived from (19) as well as (20) that

$$a_3-\mu a_2^2={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}+\left(1-\mu \right)a_2^2={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}$$

$$+{\displaystyle \frac{{\mathfrak{B}}_{q,1}^3\left(x\right) \left(u_2+v_2\right)\left(1-\mu \right)}{{\mathfrak{B}}_{q,1}^2\left(x\right)\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-\frac{2}{{\left[2\right]}_q}{\mathfrak{B}}_{q,2}\left(x\right){\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2}}$$

$$={\mathfrak{B}}_{q,1} \left(x\right)\left[\left(\phi \left(\mu ,x\right)+{\displaystyle \frac{1}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}\right)u_2+\left(\phi \left(\mu ,x\right)-{\displaystyle \frac{1}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}\right)v_2\right],$$

where

$$\phi \left(\mu ,x\right)={\displaystyle \frac{{\mathfrak{B}}_{q,1}^2\left(1-\mu \right)}{{\mathfrak{B}}_{q,1}^2\left(x\right)\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-\frac{2}{{\left[2\right]}_q}{\mathfrak{B}}_{q,2}\left(x\right){\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2}}.$$

Thus, according to (3), we have

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}} , 0\le \left|\phi \left(\mu ,x\right)\right|\le {\displaystyle \frac{1}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}\\ {\displaystyle \frac{2\left|{\left[2\right]}_{q}x-q\right|.\left|\phi \left(\mu ,x\right)\right|}{{\left[2\right]}_q}}, \left|\phi \left(\mu ,x\right)\right|\ge {\displaystyle \frac{1}{2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}. \end{array}\right.$$

After simple computation, this yields

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}} ; \\ \left|\phi -1\right|\le {\displaystyle \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right| }{2{\left({\left[2\right]}_{q}x-q\right)}^2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}},\\ {\displaystyle \frac{2{\left|{\left[2\right]}_{q}x-q\right|}^3\left|\mu -1\right|}{\begin{array}{c}{\left[2\right]}_{q }^3 \left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right|\\ \left|\phi -1\right|\ge \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)x^2\\ +2\left({\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2-q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)\left({\rm Y}+1\right)+2\mu \gimel \left(2\gimel -1\right)\right]-2{\left[\left(1-\mu \right)\left({\rm Y}+1\right)+\mu \left(2\gimel -1\right)\right]}^2\right)\end{array}\right| }{2{\left({\left[2\right]}_{q}x-q\right)}^2\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}.\end{array}}};\end{array}\right.$$

We have thus completed the proof of Theorem 3. $\square$

For $\mu ={\rm Y} =0$ in Theorem 3, given the results of Wanas and Khachi (see [5]).

Corollary 3.

[33] For $\mu \in \mathbb{R}$, let  $f\in S_{\Sigma }^{\ast }\left(q; x\right)$ . Then

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{2{\left[2\right]}_q}} ; \\ \left|\phi -1\right|\le {\displaystyle \frac{{\left[2\right]}_{q }^2\left|\left({\left[2\right]}_q-1\right)x^2+\left(1-2q\right)x+\frac{q\left(q{\left[3\right]}_q-1\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right| }{2{\left({\left[2\right]}_{q}x-q\right)}^2}},\\ {\displaystyle \frac{{\left|{\left[2\right]}_{q}x-q\right|}^3\left|\mu -1\right|}{\begin{array}{c}{\left[2\right]}_{q }^3 \left|\left({\left[2\right]}_q-1\right)x^2+\left(1-2q\right)x+\frac{q\left(q{\left[3\right]}_q-1\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right|\\ \left|\phi -1\right|\ge \frac{{\left[2\right]}_{q }^2\left|\left({\left[2\right]}_q-1\right)x^2+\left(1-2q\right)x+\frac{q\left(q{\left[3\right]}_q-1\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right| }{2{\left({\left[2\right]}_{q}x-q\right)}^2}.\end{array}}};\end{array}\right.$$

Setting $\mu =1$ in Theorem 2.3 yields the subsequent corollary.

Corollary 4.

If $f\in \mathcal{A}$ is in the family  ${\mathbb{M}}_{\Sigma }\left(1,{\rm Y},\gimel ,q; x\right)$ , then

$$\left|a_3-a_2^2\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{\left[2\right]}_q\left[\left(1-\mu \right)\left({\rm Y}+2\right)+\mu \left(3\gimel -1\right)\right]}}.$$

Theorem 4.

For  $\mu \in \mathbb{R}$ , let  $f\in \mathcal{A}$ be in the family  ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ . Then

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{3\left[2\right]}_q\left(3\mu \left(\gimel -1\right)+2\right)}}; \\ \left|\psi -1\right|\le {\displaystyle \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right| }{3{\left({\left[2\right]}_{q}x-q\right)}^2\left(3\mu \left(\gimel -1\right)+2\right)}},\\ {\displaystyle \frac{{\left|{\left[2\right]}_{q}x-q\right|}^3\left|\mu -1\right|}{\begin{array}{c}{\left[2\right]}_{q }^3 \left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right|\\ \left|\psi -1\right|\ge \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right| }{3{\left({\left[2\right]}_{q}x-q\right)}^2\left(3\mu \left(\gimel -1\right)+2\right)}.\end{array}}}\end{array}\right.;$$

Proof. 

It is derived from (32) and (33) that

$$\begin{gathered} a_3-\mu a_2^2={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{6\left(3\mu \left(\gimel -1\right)+2\right)}}+\left(1-\mu \right)a_2^2={\displaystyle \frac{{\mathfrak{B}}_{q,1}\left(x\right)\left(u_2-v_2\right)}{6\left(3\mu \left(\gimel -1\right)+2\right)}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \frac{{\mathfrak{B}}_{q,1}^3\left(x\right) \left(u_2+v_2\right)\left(1-\mu \right)}{2\left[\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]{\mathfrak{B}}_{q,1}^2\left(x\right)-\frac{4{\left(2\mu \left(\gimel -1\right)+1\right)}^2}{{\left[2\right]}_q}{\mathfrak{B}}_{q,2}\left(x\right)\right]}} \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{{\mathfrak{B}}_{q,1} \left(x\right)}{2}}\left[\left(\psi \left(\mu ,x\right)+{\displaystyle \frac{1}{3\left(3\mu \left(\gimel -1\right)+2\right)}}\right)u_2+\left(\psi \left(\mu ,x\right)-{\displaystyle \frac{1}{3\left(3\mu \left(\gimel -1\right)+2\right)}}\right)v_2\right] , \end{gathered}$$

where

$$\psi \left(\mu ,x\right)={\displaystyle \frac{{\mathfrak{B}}_{q,1}^2\left(1-\mu \right)}{\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]{\mathfrak{B}}_{q,1}^2\left(x\right)-\frac{4{\left(2\mu \left(\gimel -1\right)+1\right)}^2}{{\left[2\right]}_q}{\mathfrak{B}}_{q,2}\left(x\right)}}.$$

Thus, according to (3), we have

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{3\left[2\right]}_q\left(3\mu \left(\gimel -1\right)+2\right)}} , 0\le \left|\psi \left(\mu ,x\right)\right|\le {\displaystyle \frac{1}{3\left(3\mu \left(\gimel -1\right)+2\right)}}\\ {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|.\left|\psi \left(\mu ,x\right)\right|}{{\left[2\right]}_q}}, \left|\psi \left(\mu ,x\right)\right|\ge {\displaystyle \frac{1}{3\left(3\mu \left(\gimel -1\right)+2\right)}} . \end{array}\right.$$

which, after simple computation, yields

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{3\left[2\right]}_q\left(3\mu \left(\gimel -1\right)+2\right)}}; \\ \left|\psi -1\right|\le {\displaystyle \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right| }{3{\left({\left[2\right]}_{q}x-q\right)}^2\left(3\mu \left(\gimel -1\right)+2\right)}} ,\\ {\displaystyle \frac{{\left|{\left[2\right]}_{q}x-q\right|}^3\left|\mu -1\right|}{\begin{array}{c}{\left[2\right]}_{q }^3 \left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right|\\ \left|\psi -1\right|\ge \frac{{\left[2\right]}_{q }^2\left|\begin{array}{c}\left({\left[2\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)x^2\\ +2\left({\left(2\mu \left(\gimel -1\right)+1\right)}^2-q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]\right)x\\ +\frac{q}{{\left[2\right]}_q{\left[3\right]}_q}\left(q{\left[3\right]}_q\left[2+4{\rm Y}+9\mu (\gimel -1)+8\gimel \mu (\gimel -2)+4\mu (2-{\rm Y})\right]-4{\left(2\mu \left(\gimel -1\right)+1\right)}^2\right)\end{array}\right| }{3{\left({\left[2\right]}_{q}x-q\right)}^2\left(3\mu \left(\gimel -1\right)+2\right)} .\end{array}}}\end{array}\right.;$$

We have thus completed the proof of Theorem 4. $\square$

For $\mu ={\rm Y}=0$ in Theorem 4, given the results of Wanas and Khachi (see [5]).

Corollary 5.

[33] For $\mu \in \mathbb{R}$ , let$f\in C_{\Sigma }\left(q;x\right)$ . Then

$$\left|a_3-\mu a_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{6{\left[2\right]}_q}}; \\ \left|\psi -1\right|\le {\displaystyle \frac{{\left[2\right]}_{q }^2\left|\left({\left[2\right]}_q-2\right)x^2+2\left(1-q\right)x+\frac{q\left(q{\left[3\right]}_q-2\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right| }{3{\left({\left[2\right]}_{q}x-q\right)}^2}},\\ {\displaystyle \frac{{\left|{\left[2\right]}_{q}x-q\right|}^3\left|\mu -1\right|}{\begin{array}{c}2{\left[2\right]}_{q }^3 \left|\left({\left[2\right]}_q-2\right)x^2+2\left(1-q\right)x+\frac{q\left(q{\left[3\right]}_q-2\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right|\\ \left|\psi -1\right|\ge \frac{{\left[2\right]}_{q }^2\left|\left({\left[2\right]}_q-2\right)x^2+2\left(1-q\right)x+\frac{q\left(q{\left[3\right]}_q-2\right)}{{\left[2\right]}_q{\left[3\right]}_q}\right| }{3{\left({\left[2\right]}_{q}x-q\right)}^2}.\end{array}}}\end{array}\right.;$$

Setting $\mu =1$ in Theorem 4 yields the subsequent corollary.

Corollary 6.

If  $f\in \mathcal{A}$ is in the family  ${\mathbb{N}}_{\Sigma }\left(1,{\rm Y},\gimel ,q; x\right)$, then

$$\left|a_3-a_2^2\right|\le {\displaystyle \frac{\left|{\left[2\right]}_{q}x-q\right|}{{3\left[2\right]}_q\left(3\mu \left(\gimel -1\right)+2\right)}}.$$

3. Example and Applications

Consider the function

$$f\left(\mathbb{ᶎ}\right)={\displaystyle \frac{\mathbb{ᶎ}}{1-\mathbb{ᶎ}}}=\mathbb{ᶎ}+{\mathbb{ᶎ}}^2+{\mathbb{ᶎ}}^3+\cdots , \mathbb{ᶎ}\in U$$

which belongs to the family $\Sigma$ with the inverse function $g\left(w\right)=={\displaystyle \frac{w}{1+w}}$. Substituting into the subordinations of Definitions 1 and 2, it is straightforward to verify that $f$ belongs to the family ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ for suitable parameter choices. Hence, the coefficient bounds obtained in Theorems 1–3 can be applied directly to this function.

As an application, our results provide estimates for the initial coefficients $a_2$,$a_3$ of such bi-univalent functions and yield bounds for the Fekete–Szegö functional

$$\left|a_3-\mu a_2^2\right|.$$

These estimates play an important role in Geometric Function Theory, particularly in understanding the growth, distortion, and covering properties of analytic and bi-univalent functions. Furthermore, since the newly introduced subclasses are connected with Bernoulli polynomials, the results may also have implications in approximation theory and numerical analysis, where Bernoulli polynomials naturally arise.

4. Conclusions

The diverse and effective applications of several intriguing special polynomials in the Geometric Function Theory of Complex Investigation served as the principal inspiration and impetus for our investigation in this work. It is noteworthy that numerous recent studies addressing certain problems of our presentation in this study extensively utilized basic or quantum (q-) calculus (see, for instance, [34,35]). Our main objective was to define certain families, ${\mathbb{M}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$ and ${\mathbb{N}}_{\Sigma }\left(\mu ,{\rm Y},\gimel ,q; x\right)$, of Bazilevič and $\gimel$-pseudo bi-univalent holomorphic functions, which are defined by means of the $q$-Bernoulli polynomial ${\mathfrak{B}}_{q,n}(x)$ given by the recurrence relation (3) and by generating function ${\mathfrak{B}}_q(x,h)$ in (4). We have investigated inequalities for the initial Taylor–Maclaurin coefficients and Fekete- Szegö problem of functions belonging to these introduced families.

This article is based on a recently published survey and explanatory study by Srivastava [25], which investigated the mathematical applications of $q$-calculus, fractional $q$-calculus, particularly fractional q-derivative operators in the context of the Geometric Function Theory of Complex Analysis, particularly concerning the Fekete–Szegö functional. Srivastava [25] elucidated the not-yet-commonly comprehended notion that the purported ($p,q$)-variation in classical $q$-calculus is, in reality, a rather trivial and inconsequential modification of classical $q$-calculus, with the additional parameter $p$ being redundant or superfluous (refer to [25] (p. 340); also see [25] (pp. 1511–1512)).

Regarding future research directions, the contents of this paper on a $q$-Bernoulli polynomial could inspire further research related to other families, and the symmetry properties of this newly introduced operator can be studied.