A Generalization of Gegenbauer Polynomials and Bi-Univalent Functions

Abstract

Three subclasses of analytic and bi-univalent functions are introduced through the use of $q-$Gegenbauer polynomials, which are a generalization of Gegenbauer polynomials. For functions falling within these subclasses, coefficient bounds $\left|a_2\right|$ and $\left|a_3\right|$ as well as Fekete–Szegö inequalities are derived. Specializing the parameters used in our main results leads to a number of new results.

1. Introduction

Legendre first made the discovery of orthogonal polynomials in 1784 [1]. Under specific model restrictions, orthogonal polynomials are frequently employed to solve ordinary differential equations. Furthermore, a crucial function in the approximation theory is performed by orthogonal polynomials [2].

$P_m$ and $P_n$ are two polynomials of order m and n, respectively, and are orthogonal if

$${\int }_a^b{P}_n\left(x\right)P_m\left(x\right)s\left(x\right)dx=0,\mathrm{for}m\ne n,$$

where $s\left(x\right)$ is a suitably specified function in the interval $(a,b)$; therefore, all finite order polynomials $P_n\left(x\right)$ have a well-defined integral.

Gegenbauer polynomials are orthogonal polynomials of a specified type. As found in [3], when traditional algebraic formulations are used, the generating function of Gegenbauer polynomials and the integral representation of typically real functions $T_R$ are related to each other in a symbolic way $T_R$. This undoubtedly caused a number of helpful inequalities to emerge from the world of Gegenbauer polynomials.

$q-$orthogonal polynomials are now of particular relevance in both physics and mathematics due to the development of quantum groups. The $q-$deformed harmonic oscillator, for instance, has a group-theoretic setting for the $q-$Laguerre and $q-$Hermite polynomials. Jackson’s $q-$exponential plays a crucial role in the mathematical framework required to characterize the properties of these $q-$polynomials, such as the recurrence relations, generating functions, and orthogonality relations. Jackson’s $q-$exponential has recently been expressed by Quesne [4] as a closed-form multiplicative series of regular exponentials with known coefficients. In this case, it is crucial to look into how this discovery might affect the theory of $q-$orthogonal polynomials. An effort in this regard was made in the current work. To obtain novel nonlinear connection equations for $q-$Gegenbauer polynomials in terms of their respective classical equivalents, we used the aforementioned result in particular.

This study analyzed various features of the class under consideration after associating some bi-univalent functions with $q-$Gegenbauer polynomials. The following part lays the foundation for mathematical notations and definitions.

2. Preliminaries

Let $\mathcal{A}$ denote the class of all analytical functions f that are defined on the open unit disk $\mathbb{U}=\{\xi \in \mathbb{C}:\left|\xi \right|<1\}$ and normalized by the formula $f\left(0\right)=$ $f^{\prime }\left(0\right)-1=0$. As a result, each $f\in \mathcal{A}$ has the following Taylor–Maclaurin series expansion:

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

(1)

In addition, let $\mathcal{S}$ denote the class of all functions $f\in \mathcal{A}$ that are univalent in $\mathbb{U}$.

Let the functions $g\left(\xi \right)$ and $f\left(\xi \right)$ be analytic in $\mathbb{U}$. We say that the function $f\left(\xi \right)$ is subordinate to $g\left(\xi \right)$, written as $f\left(\xi \right)\prec g\left(\xi \right)$, if there exists a Schwarz function $\omega$ that is analytic in $\mathbb{U}$ with

$$\left|\omega \right(\xi \left)\right|<1\mathrm{and}\omega \left(0\right)=0(\xi \in \mathbb{U})$$

such that

$$g\left(\omega \right(\xi \left)\right)=f\left(\xi \right).$$

Beside that, if the function g is univalent in $\mathbb{U}$, then the following equivalence holds:

$$f\left(\xi \right)\prec g\left(\xi \right)\mathrm{if}g\left(0\right)=f\left(0\right)$$

and

$$f\left(\mathbb{U}\right)\subset g\left(\mathbb{U}\right).$$

It is well known that every function $f\in \mathcal{S}$ has an inverse $f^{-1}$, defined by

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

and

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

where

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

(2)

If both $f^{-1}\left(\xi \right)$ and $f\left(\xi \right)$ are univalent in $\mathbb{U}$, then a function is said to be bi-univalent in $\mathbb{U}$.

Let $\Sigma$ denote the class of bi-univalent functions in $\mathbb{U}$ given by (1). Examples of functions in the class $\Sigma$ are $\frac{\xi }{1-\xi },$$log\sqrt{\frac{1+\xi }{1-\xi }}$.

Fekete and Szegö achieved a sharp bound of the functional $\eta a_2^2-a_3$, with real $\eta$$(0\le \eta \le 1)$ for a univalent function f in 1933 [5]. Since that time, it has been known as the classical Fekete and Szegö problem of establishing the sharp bounds for this functional of any compact family of functions f$\in \mathcal{A}$ with any complex $\eta$.

In 1983, Askey and Ismail [6] found a class of polynomials that can be interpreted as q–analogues of the Gegenbauer polynomials. These are essentially the polynomials ${\mathfrak{B}}_q^{\left(\lambda \right)}(\xi ,z)$

$${\mathfrak{G}}_q^{\left(\lambda \right)}(x,\xi )=\sum _{n=0}^{\infty }{\mathcal{C}}_n^{\left(\lambda \right)}(x;q){\xi }^n,$$

(3)

where $x\in [-1,1]$ and $\xi \in \mathbb{U}$.

In 2006, Chakrabarti et al. [7] found a class of polynomials that can be interpreted as q–analogues of the Gegenbauer polynomials by the following recurrence relations:

$$\begin{array}{cc}\hfill {\mathcal{C}}_0^{\left(\lambda \right)}(x;q)& =1,{\mathcal{C}}_1^{\left(\lambda \right)}(x;q)={\left[\lambda \right]}_q{C}_1^1\left(x\right)=2{\left[\lambda \right]}_{q}x,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill {\mathcal{C}}_2^{\left(\lambda \right)}(x;q)& ={\left[\lambda \right]}_{q^2}{C}_2^1\left(x\right)-\frac{1}{2}\left({\left[\lambda \right]}_{q^2}-{\left[\lambda \right]}_q^2\right)C_1^2\left(x\right)=2\left({\left[\lambda \right]}_{q^2}+{\left[\lambda \right]}_q^2\right)x^2-{\left[\lambda \right]}_{q^2}.\hfill \end{array}$$

(5)

where $0<q<1$ and $\lambda \in \mathbb{N}=\left\{1,2,3,\cdots \right\}.$

In 2021, Amourah et al. [8,9] considered the classical Gegenbauer polynomials ${\mathfrak{G}}^{\left(\lambda \right)}(x,\xi )$, where $\xi \in \mathbb{U}$ and $x\in [-1,1]$. For fixed x, the function ${\mathfrak{G}}^{\left(\lambda \right)}$ is analytic in $\mathbb{U}$, so it can be expanded in a Taylor series as

$${\mathfrak{G}}^{\left(\lambda \right)}(x,\xi )=\sum _{n=0}^{\infty }{C}_n^{\alpha }\left(x\right){\xi }^n,$$

where $C_n^{\alpha }\left(x\right)$ is the classical Gegenbauer polynomial of degree n.

Recently, several authors have begun examining bi-univalent functions connected to orthogonal polynomials (such as [10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28]).

As far as we are aware, there is no published work on bi-univalent functions for $q-$Gegenbauer polynomials. The major objective of this work is to start an investigation of the characteristics of bi-univalent functions related to $q-$Gegenbauer polynomials. To perform this, we consider the following definitions in the next section.

3. Coefficient Bounds of the Class ${\mathfrak{B}}_{\Sigma }(x,\alpha ;q)$

Here, we introduce some new bi-univalent function subclasses that are subordinate to the $q-$Gegenbauer polynomial.

Definition 1. 

For $x\in (\frac{1}{2},1]$ and $0<q<1$, if the following subordinations are satisfied, a function f belonging to Σ is said to be in the class ${\mathfrak{B}}_{\Sigma }(x,\alpha ;q)$ given by (1):

$${\partial }_{q}f\left(\xi \right)\prec {\mathfrak{G}}_q^{\left(\lambda \right)}(x,\xi )$$

(6)

and

$${\partial }_{q}g\left(w\right)\prec {\mathfrak{G}}_q^{\left(\lambda \right)}(x,w),$$

(7)

where $\lambda \in \mathbb{N}=\left\{1,2,3,\cdots \right\}$, $\alpha$ is a nonzero real constant, the function $f^{-1}\left(w\right)=g\left(w\right)$ is defined by (2), and ${\mathfrak{G}}_q^{\left(\lambda \right)}$ is the generating function of q–analogues of the Gegenbauer polynomials given by (3).

We start by providing the coefficient estimates for the class ${\mathfrak{B}}_{\Sigma }(x,\alpha ;q)$ specified in Definition 1.

Theorem 1. 

Let $f\in \Sigma$ given by (1) be in the class ${\mathfrak{B}}_{\Sigma }(x,\alpha ;q).$ Then,

$$\left|a_2\right|\le \frac{2x\left|{\left[\lambda \right]}_q\right|\sqrt{2{\left[\lambda \right]}_{q}x}}{{\left[2\right]}_q\sqrt{\left[\left(\frac{4{\left[3\right]}_q}{{\left[2\right]}_q^2}-2\right){\left[\lambda \right]}_q^2-2{\left[\lambda \right]}_{q^2}\right]x^2+{\left[\lambda \right]}_q}},$$

and

$$\left|a_3\right|\le \frac{4{\left[\lambda \right]}_q^2{x}^2}{{\left[2\right]}_q^2}+\frac{2\left|{\left[\lambda \right]}_q\right|x}{{\left[3\right]}_q}.$$

Proof. 

Let $f\in {\mathfrak{B}}_{\Sigma }(x,\alpha ;q)$. From Definition 1, for some analytic functions w and v such that $w\left(0\right)=0=v\left(0\right)$ and $\left|w\right(\xi \left)\right|<1$, $\left|v\right(w\left)\right|<1$ for all $w,\xi \in \mathbb{U}$; then, we can write

$${\partial }_{q}f\left(\xi \right)={\mathfrak{G}}_q^{\left(\lambda \right)}(x,w\left(\xi \right))$$

(8)

and

$${\partial }_{q}g\left(w\right)={\mathfrak{G}}_q^{\left(\lambda \right)}(x,v\left(w\right)),$$

(9)

From the Equations (8) and (9), we obtain that

$${\partial }_{q}f\left(\xi \right)=1+C_1^{\left(\lambda \right)}(x;q)c_1\xi +\left[C_1^{\left(\lambda \right)}(x;q)c_2+C_2^{\left(\lambda \right)}(x;q)c_1^2\right]{\xi }^2+\cdots$$

(10)

and

$${\partial }_{q}g\left(w\right)=1+C_1^{\left(\lambda \right)}(x;q)d_{1}w+\left[C_1^{\left(\lambda \right)}(x;q)d_2+C_2^{\left(\lambda \right)}(x;q)d_1^2\right])w^2+\cdots .$$

(11)

It is generally understood that if

$$\left|w\left(\xi \right)\right|=\left|c_1\xi +c_2{\xi }^2+c_3{\xi }^3+\cdots \right|<1,(\xi \in \mathbb{U})$$

and

$$\left|v\left(w\right)\right|=\left|d_{1}w+d_2{w}^2+d_3{w}^3+\cdots \right|<1,(w\in \mathbb{U}),$$

then

$$|c_j|\le 1\mathrm{and}|d_j|\le 1\mathrm{for}\mathrm{all}j\in \mathbb{N}.$$

(12)

As a result, we have the following after comparing the relevant coefficients in (10) and (11):

$${\left[2\right]}_q{a}_2=C_1^{\left(\lambda \right)}(x;q)c_1,$$

(13)

$${\left[3\right]}_q{a}_3=C_1^{\left(\lambda \right)}(x;q)c_2+C_2^{\left(\lambda \right)}(x;q)c_1^2,$$

(14)

$$-{\left[2\right]}_q{a}_2=C_1^{\left(\lambda \right)}(x;q)d_1,$$

(15)

and

$${\left[3\right]}_q\left(2a_2^2-a_3\right)=C_1^{\left(\lambda \right)}(x;q)d_2+C_2^{\left(\lambda \right)}(x;q)d_1^2.$$

(16)

From the Equations (13) and (15), we have

$$c_1=-d_1$$

(17)

and

$$2{\left[2\right]}_q^2{a}_2^2={\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2\left(c_1^2+d_1^2\right).$$

(18)

By adding (14) to (16), yields

$$2{\left[3\right]}_q{a}_2^2=C_1^{\left(\lambda \right)}(x;q)\left(c_2+d_2\right)+C_2^{\left(\lambda \right)}(x;q)\left(c_1^2+d_1^2\right).$$

(19)

We determine that, by replacing the value of $\left(c_1^2+d_1^2\right)$ from (18) on the right side of (19),

$$\left(2{\left[3\right]}_q-\frac{2C_2^{\left(\lambda \right)}(x;q){\left[2\right]}_q^2}{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2}\right)a_2^2=C_1^{\left(\lambda \right)}(x;q)\left(c_2+d_2\right).$$

(20)

Through computations using (11), (5) and (20), we find that

$$\left|a_2\right|\le \frac{2\left|{\left[\lambda \right]}_q\right|x\sqrt{2{\left[\lambda \right]}_{q}x}}{{\left[2\right]}_q\sqrt{\left[\left(\frac{4{\left[3\right]}_q}{{\left[2\right]}_q^2}-2\right){\left[\lambda \right]}_q^2-2{\left[\lambda \right]}_{q^2}\right]x^2+{\left[\lambda \right]}_q}}.$$

In addition, if we subtract (16) from (14), we obtain

$$2{\left[3\right]}_q\left(a_3-a_2^2\right)=C_1^{\left(\lambda \right)}(x;q)\left(c_2-d_2\right)+C_2^{\left(\lambda \right)}(x;q)\left(c_1^2-d_1^2\right).$$

(21)

Then, in view of (18) and (21), we obtain

$$a_3=\frac{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2}{2{\left[2\right]}_q^2}\left(c_1^2+d_1^2\right)+\frac{C_1^{\left(\lambda \right)}(x;q)}{2{\left[3\right]}_q}\left(c_2-d_2\right).$$

By applying (4), we conclude that

$$\left|a_3\right|\le \frac{4{\left[\lambda \right]}_q^2{x}^2}{{\left[2\right]}_q^2}+\frac{2\left|{\left[\lambda \right]}_q\right|x}{{\left[3\right]}_q}.$$

The proof of the theorem is now complete. □

Using the values of $a_2^2$ and $a_3$, we prove the following Fekete–Szegö inequality for functions in the class ${\mathfrak{B}}_{\Sigma }(x,\alpha ;q)$.

Theorem 2. 

Let $f\in \Sigma$ given by (1) be in the class ${\mathfrak{B}}_{\Sigma }(x,\alpha ;q)$. Then,

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{2x\left|{\left[\lambda \right]}_q\right|}{{\left[3\right]}_q},\hfill & |\sigma -1|\le \left|\frac{\left(\left(2{\left[3\right]}_q-{\left[2\right]}_q^2\right){\left[\lambda \right]}_q^2-{\left[2\right]}_q^2{\left[\lambda \right]}_{q^2}\right)x^2+2{\left[2\right]}_q^2{\left[\lambda \right]}_{q^2}}{2{\left[3\right]}_q{\left[\lambda \right]}_q^2{x}^2}\right|,\hfill \\ & \\ 2\left|2{\left[\lambda \right]}_{q}x\right|\left|h\left(\eta \right)\right|,\hfill & |\sigma -1|\ge \left|\frac{\left(\left(2{\left[3\right]}_q-{\left[2\right]}_q^2\right){\left[\lambda \right]}_q^2-{\left[2\right]}_q^2{\left[\lambda \right]}_{q^2}\right)x^2+2{\left[2\right]}_q^2{\left[\lambda \right]}_{q^2}}{2{\left[3\right]}_q{\left[\lambda \right]}_q^2{x}^2}\right|.\hfill \end{array}\right.$$

Proof. 

From (20) and (21),

$$\begin{array}{cc}\hfill a_3-\sigma a_2^2& =\left(1-\sigma \right)\frac{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^3\left(c_2+d_2\right)}{\left[2{\left[3\right]}_q{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2-2{\left[2\right]}_q^2{C}_2^{\left(\lambda \right)}(x;q)\right]}+\frac{C_1^{\left(\lambda \right)}(x;q)}{2{\left[3\right]}_q}\left(c_2-d_2\right)\hfill \\ \hfill & =C_1^{\alpha }\left(x\right)\left[\left[h\left(\eta \right)+\frac{1}{2{\left[3\right]}_q}\right]c_2+\left[h\left(\eta \right)-\frac{1}{2{\left[3\right]}_q}\right]d_2\right],\hfill \end{array}$$

where

$$\mathcal{K}\left(\sigma \right)=\frac{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2\left(1-\sigma \right)}{2{\left[3\right]}_q{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2-2{\left[2\right]}_q^2{C}_2^{\left(\lambda \right)}(x;q)},$$

In view of (4) and (5), we conclude that

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{\left|C_1^{\left(\lambda \right)}(x;q)\right|}{{\left[3\right]}_q},\hfill & \left|\mathcal{K}\left(\sigma \right)\right|\le \frac{1}{2{\left[3\right]}_q},\hfill \\ & \\ 2\left|C_1^{\left(\lambda \right)}(x;q)\right|\left|\mathcal{K}\left(\sigma \right)\right|,\hfill & \left|\mathcal{K}\left(\sigma \right)\right|\ge \frac{1}{2{\left[3\right]}_q}.\hfill \end{array}\right.$$

The proof of the theorem is now complete. □

Corollary 1. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathfrak{B}}_{\Sigma }(x,\alpha ;1)$. Then,

$$\left|a_2\right|\le \frac{\left|\alpha \right|x\sqrt{2\left|\alpha \right|x}}{\sqrt{\left|\left[\left(\alpha -2\right)x^2+1\right]\alpha \right|}}.$$

$$\left|a_3\right|\le {\lambda }^2{x}^2+\frac{2\left|\lambda \right|x}{3},$$

and $\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2\left|\alpha \right|x}{3},\\ \\ \frac{2{\left|\alpha x\right|}^3\left|1-\eta \right|}{\left|\left[\left(\alpha -2\right)x^2+1\right]\alpha \right|},\end{array}\right.\begin{array}{c}\left|\eta -1\right|\le \left|\frac{\left(\alpha -2\right)x^2+1}{3\alpha x^2}\right|\\ \\ \left|\eta -1\right|\ge \left|\frac{\left(\alpha -2\right)x^2+1}{3\alpha x^2}\right|.\end{array}$

Corollary 2. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathfrak{B}}_{\Sigma }(x,1;1)$. Then,

$$\left|a_2\right|\le \frac{x\sqrt{2x}}{\sqrt{\left|1-x^2\right|}},$$

$$\left|a_3\right|\le x^2+\frac{2x}{3},$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{c}\frac{2x}{3},\\ \\ \frac{2x^3\left|1-\eta \right|}{\left|1-x^2\right|},\end{array}\right.\begin{array}{c}\left|\eta -1\right|\le \left|\frac{1-x^2}{3x^2}\right|\\ \\ \left|\eta -1\right|\ge \left|\frac{1-x^2}{3x^2}\right|.\end{array}$$

4. Coefficient Bounds of the Class ${\mathcal{S}}_{\Sigma }^{*}(x,\alpha ;q)$

Definition 2. 

For $x\in (\frac{1}{2},1]$ and $0<q<1$, if the following subordinations are satisfied, a function f belonging to Σ is said to be in the class ${\mathcal{S}}_{\Sigma }^{*}(x,\alpha ;q)$ given by (1):

$$\frac{\xi {\partial }_{q}f\left(\xi \right)}{f\left(\xi \right)}\prec {\mathfrak{G}}_q^{\left(\lambda \right)}(x,\xi ),$$

(22)

and

$$\frac{w{\partial }_{q}g\left(w\right)}{g\left(w\right)}\prec {\mathfrak{G}}_q^{\left(\lambda \right)}(x,w),$$

(23)

where $\lambda \in \mathbb{N}=\left\{1,2,3,\cdots \right\}$, $\alpha$ is a nonzero real constant, the function $g\left(w\right)=f^{-1}\left(w\right)$ is defined by (2), and ${\mathfrak{G}}_q^{\left(\lambda \right)}$ is the generating function of the q–analogues of Gegenbauer polynomials given by (3).

Theorem 3. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathcal{S}}_{\Sigma }^{*}(x,\alpha ;q)$. Then, we have

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

and

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

Proof. 

Let $f\in {\mathcal{S}}_{\Sigma }^{*}(x,\alpha ;q)$. From Definition 2, for some analytic functions w and v such that $w\left(0\right)=0=v\left(0\right)$ and $\left|w\right(\xi \left)\right|<1,$$\left|v\right(w\left)\right|<1$ for all $\xi ,w\in \mathbb{U}$,

$$\frac{\xi {\partial }_{q}f\left(\xi \right)}{f\left(\xi \right)}={\mathfrak{G}}_q^{\left(\lambda \right)}(x,w\left(\xi \right)),$$

(24)

and

$$\frac{\xi {\partial }_{q}g\left(w\right)}{g\left(w\right)}={\mathfrak{G}}_q^{\left(\lambda \right)}(x,v\left(w\right)).$$

(25)

From the equalities (24) and (25), we obtain that

$$\frac{\xi {\partial }_{q}f\left(\xi \right)}{f\left(\xi \right)}=1+C_1^{\left(\lambda \right)}(x;q)\xi +\left[C_1^{\left(\lambda \right)}(x;q)c_2+C_2^{\left(\lambda \right)}(x;q)c_1^2\right]{\xi }^2+\cdots$$

(26)

and

$$\frac{\xi {\partial }_{q}g\left(w\right)}{g\left(w\right)}=1+C_1^{\left(\lambda \right)}(x;q)d_{1}w+\left[C_1^{\left(\lambda \right)}(x;q)d_2+C_2^{\left(\lambda \right)}(x;q)d_1^2\right])w^2+\cdots .$$

(27)

Thus, upon comparing the corresponding coefficients in (26) and (27), we have

$$q{a}_2=C_1^{\left(\lambda \right)}(x;q)c_1,$$

(28)

$$q\left(1+q\right)a_3-q{a}_2^2=C_1^{\left(\lambda \right)}(x;q)c_2+C_2^{\left(\lambda \right)}(x;q)c_1^2,$$

(29)

$$-q{a}_2=C_1^{\left(\lambda \right)}(x;q)d_1,$$

(30)

and

$$q\left(1+2q\right)a_2^2-q\left(1+q\right)a_3=C_1^{\left(\lambda \right)}(x;q)d_2+C_2^{\left(\lambda \right)}(x;q)d_1^2.$$

(31)

From the Equations (28) and (30), it follows that

$$c_1=-d_1$$

(32)

and

$$2q^2{a}_2^2={\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2\left(c_1^2+d_1^2\right).$$

(33)

By adding (29) to (31), yields

$$2q^2{a}_2^2=C_1^{\left(\lambda \right)}(x;q)\left(c_2+d_2\right)+C_2^{\left(\lambda \right)}(x;q)\left(c_1^2+d_1^2\right).$$

(34)

We determine that, by replacing the value of $\left(c_1^2+d_1^2\right)$ from (33) on the right side of (34),

$$2q^2\left(1-\frac{C_2^{\left(\lambda \right)}(x;q)}{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2}\right)a_2^2=C_1^{\left(\lambda \right)}(x;q)\left(c_2+d_2\right).$$

(35)

Moreover, through computations using (5) and (35), we find that

$$|a_2|\le \frac{2\left|{\left[\lambda \right]}_q\right|x\sqrt{2{\left[\lambda \right]}_{q}x}}{q\sqrt{2\left({\left[\lambda \right]}_q^2-{\left[\lambda \right]}_{q^2}\right)x^2+{\left[\lambda \right]}_{q^2}}}.$$

Now, if we subtract (31) from (29), we obtain

$$2q\left(1+q\right)\left(a_3-a_2^2\right)=C_1^{\left(\lambda \right)}(x;q)\left(c_2-d_2\right)+C_2^{\left(\lambda \right)}(x;q)\left(c_1^2-d_1^2\right).$$

(36)

By viewing of (33) and (36), we conclude that

$$a_3=\frac{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2}{2q^2}\left(c_1^2+d_1^2\right).+\frac{C_1^{\left(\lambda \right)}(x;q)}{2q\left(1+q\right)}\left(c_2-d_2\right).$$

By applying (4) and (5), we have

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

This completes the proof of the Theorem 3. □

Theorem 4. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathcal{S}}_{\Sigma }^{*}(x,\alpha ;q)$. Then,

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{\left|{\left[\lambda \right]}_q\right|x}{q(1+q)},\hfill & \left|1-\sigma \right|\le \frac{q^2\left[\left(2\left({\left[\lambda \right]}_q^2-{\left[\lambda \right]}_{q^2}\right)x^2\right)+{\left[\lambda \right]}_{q^2}\right]}{8(1+q)|{\left[\lambda \right]}_{q}x{|}^3},\hfill \\ & \\ \frac{8|{\left[\lambda \right]}_{q}x{|}^3\left|1-\sigma \right|}{q^2\left[\left(2\left({\left[\lambda \right]}_q^2-{\left[\lambda \right]}_{q^2}\right)x^2\right)+{\left[\lambda \right]}_{q^2}\right]},\hfill & \left|1-\sigma \right|\ge \frac{q^2\left[\left(2\left({\left[\lambda \right]}_q^2-{\left[\lambda \right]}_{q^2}\right)x^2\right)+{\left[\lambda \right]}_{q^2}\right]}{8(1+q)|{\left[\lambda \right]}_{q}x{|}^3}.\hfill \end{array}\right.$$

Proof. 

From (35) and (36),

$$\begin{array}{cc}\hfill a_3-\sigma a_2^2& =\frac{(1-\sigma ){\left(C_1^{\left(\lambda \right)}(x;q)\right)}^3}{2q^2\left({\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2-C_2^{\left(\lambda \right)}(x;q)\right)}\left(c_2+d_2\right)+\frac{C_1^{\left(\lambda \right)}(x;q)}{2q\left(1+q\right)}\left(c_2-d_2\right)\hfill \\ \hfill & =\frac{C_1^{\left(\lambda \right)}(x;q)}{2q}\left[\left[\mathfrak{K}\left(\sigma \right)+\frac{1}{1+q}\right]c_2+\left[\mathfrak{K}\left(\sigma \right)-\frac{1}{1+q}\right]d_2\right],\hfill \end{array}$$

where

$$\mathfrak{K}\left(\sigma \right)=\frac{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2\left(1-\sigma \right)}{q\left[{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2-C_2^{\left(\lambda \right)}(x;q)\right]},$$

$$\left|1-\sigma \right|\ge \frac{q^2\left[\left(2\left({\left[\lambda \right]}_q^2-{\left[\lambda \right]}_{q^2}\right)x^2\right)+{\left[\lambda \right]}_{q^2}\right]}{8(1+q)|{\left[\lambda \right]}_{q}x{|}^3},$$

then, in view of (4) and (5), we conclude that

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{\left|C_1^{\left(\lambda \right)}(x;q)\right|}{2q(1+q)},\hfill & \left|\mathfrak{K}\left(\sigma \right)\right|\le \frac{1}{1+q},\hfill \\ & \\ \frac{1}{q}\left|C_1^{\left(\lambda \right)}(x;q)\right|\left|\mathfrak{K}\left(\sigma \right)\right|,\hfill & \left|\mathfrak{K}\left(\sigma \right)\right|\ge \frac{1}{1+q}.\hfill \end{array}\right.$$

This completes the proof of the Theorem 4. □

Corollary 3. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathcal{S}}_{\Sigma }^{*}(x,\alpha ;1)$. Then, we have

$$|a_2|\le \frac{2\left|\lambda \right|x\sqrt{2x}}{\sqrt{2\left(\lambda -1\right)x^2-1}},\left|a_3\right|\le \lambda \left(4\lambda x+1\right),$$

and

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{\left|\lambda \right|x}{2},\hfill & \left|1-\sigma \right|\le \frac{\left(2\left(\lambda -1\right)x^2\right)+1}{{16\left|\lambda \right|}^2{x}^3},\hfill \\ & \\ \frac{8{\lambda }^2{x}^3\left|1-\sigma \right|}{\left(2\left(\lambda -1\right)x^2\right)+1},\hfill & \left|1-\sigma \right|\ge \frac{\left(2\left(\lambda -1\right)x^2\right)+1}{{16\left|\lambda \right|}^2{x}^3}.\hfill \end{array}\right.$$

5. Coefficient Bounds of the Class ${\mathfrak{C}}_{\Sigma }(x,\alpha ;q)$

Definition 3. 

For $x\in (\frac{1}{2},1]$ and $0<q<1$, if the following subordinations are satisfied, a function f belonging to Σ is said to be in the class ${\mathfrak{C}}_{\Sigma }(x,\alpha ;q)$ given by (1):

$$1+\frac{\xi {\partial }_q^{2}f\left(\xi \right)}{{\partial }_{q}f\left(\xi \right)}\prec {\mathfrak{G}}_q^{\left(\lambda \right)}(x,\xi )$$

(37)

and

$$1+\frac{\xi {\partial }_q^{2}g\left(w\right)}{{\partial }_{q}g\left(w\right)}\prec {\mathfrak{G}}_q^{\left(\lambda \right)}(x,w),$$

(38)

where $\lambda \in \mathbb{N}=\left\{1,2,3,\cdots \right\}$, $\alpha$ is a nonzero real constant, the function $g\left(w\right)=f^{-1}\left(w\right)$ is defined by (2), and ${\mathfrak{G}}_q^{\left(\lambda \right)}$ is the generating function of the q–analogues of Gegenbauer polynomials given by (3).

Theorem 5. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathfrak{C}}_{\Sigma }(x,\alpha ;q)$. Then,

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

and

$$|a_3|\le \frac{4{\left[\lambda \right]}_q^2{x}^2}{{\left[2\right]}_q^2}+\frac{2{\left[\lambda \right]}_{q}x}{{\left[2\right]}_q{\left[3\right]}_q}.$$

Proof. 

Let $f\in {\mathfrak{C}}_{\Sigma }(x,\alpha ,\mu ;q)$. From Definition 3, for some analytic functions $w,$ v such that $w\left(0\right)=v\left(0\right)=0$ and $\left|w\right(\xi \left)\right|<1,$$\left|v\right(w\left)\right|<1$ for all $\xi ,w\in \mathbb{U}$,

$$1+\frac{\xi {\partial }_q^{2}f\left(\xi \right)}{{\partial }_{q}f\left(\xi \right)}={\mathfrak{G}}_q^{\left(\lambda \right)}(x,w\left(\xi \right)),$$

(39)

and

$$1+\frac{\xi {\partial }_q^{2}g\left(w\right)}{{\partial }_{q}g\left(w\right)}={\mathfrak{G}}_q^{\left(\lambda \right)}(x,v\left(w\right)).$$

(40)

By expanding the Equations (39) and (40), we obtain that

$$1+\frac{\xi {\partial }_q^{2}f\left(\xi \right)}{{\partial }_{q}f\left(\xi \right)}=1+C_1^{\left(\lambda \right)}(x;q)c_1\xi +\left[C_1^{\left(\lambda \right)}(x;q)c_2+C_2^{\left(\lambda \right)}(x;q)c_1^2\right]{\xi }^2+\cdots$$

(41)

and

$$1+\frac{\xi {\partial }_q^{2}g\left(w\right)}{{\partial }_{q}g\left(w\right)}=1+C_1^{\left(\lambda \right)}(x;q)d_{1}w+\left[C_1^{\left(\lambda \right)}(x;q)d_2+C_2^{\left(\lambda \right)}(x;q)d_1^2\right])w^2+\cdots .$$

(42)

Upon comparing the corresponding coefficients in (41) and (42), we have

$${\left[2\right]}_q{a}_2=C_1^{\left(\lambda \right)}(x;q)c_1,$$

(43)

$${\left[2\right]}_q{\left[3\right]}_q{a}_3-{\left[2\right]}_q^2{a}_2^2=C_1^{\left(\lambda \right)}(x;q)c_2+C_2^{\left(\lambda \right)}(x;q)c_1^2,$$

(44)

$$-{\left[2\right]}_q{a}_2=C_1^{\left(\lambda \right)}(x;q)d_1,$$

(45)

and

$${\left[2\right]}_q\left(2{\left[3\right]}_q-{\left[2\right]}_q\right)a_2^2-{\left[2\right]}_q{\left[3\right]}_q{a}_3=C_1^{\left(\lambda \right)}(x;q)d_2+C_2^{\left(\lambda \right)}(x;q)d_1^2.$$

(46)

We get from (43) and (45) that

$$c_1=-d_1$$

(47)

and

$$2{\left({\left[2\right]}_q\right)}^2{a}_2^2={\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2\left(c_1^2+d_1^2\right).$$

(48)

By adding (44) to (46), we obtain

$$2{\left[2\right]}_q\left({\left[3\right]}_q-{\left[2\right]}_q\right)a_2^2=C_1^{\left(\lambda \right)}(x;q)\left(c_2+d_2\right)+C_2^{\left(\lambda \right)}(x;q)\left(c_1^2+d_1^2\right).$$

(49)

We determine that, by replacing the value of $\left(c_1^2+d_1^2\right)$ from (48) on the right side of (49),

$$2{\left[2\right]}_q\left(\left({\left[3\right]}_q-{\left[2\right]}_q\right)-{\left[2\right]}_q\frac{C_2^{\left(\lambda \right)}(x;q)}{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2}\right)a_2^2=C_1^{\left(\lambda \right)}(x;q)\left(c_2+d_2\right).$$

(50)

Moreover, by doing computations along (12) and (50), we find that

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

By subtracting (44) from (46), we obtain

$$2{\left[2\right]}_q{\left[3\right]}_q\left(a_3-a_2^2\right)=C_1^{\left(\lambda \right)}(x;q)\left(c_2-d_2\right)+C_2^{\left(\lambda \right)}(x;q)\left(c_1^2-d_1^2\right).$$

(51)

In view of (48) and (51), we obtain

$$a_3=\frac{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2}{2{\left({\left[2\right]}_q\right)}^2}\left(c_1^2+d_1^2\right)+\frac{C_1^{\left(\lambda \right)}(x;q)}{2{\left[2\right]}_q{\left[3\right]}_q}\left(c_2-d_2\right).$$

By applying (5), we conclude that

$$|a_3|\le \frac{4{\left[\lambda \right]}_q^2{x}^2}{{\left[2\right]}_q^2}+\frac{2{\left[\lambda \right]}_{q}x}{{\left[2\right]}_q{\left[3\right]}_q}.$$

This completes the proof of the Theorem 5. □

Theorem 6. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathfrak{C}}_{\Sigma }(x,\alpha ;q)$. Then,

$$|a_3-\sigma a_2^2|\le$$

$$\left\{\begin{array}{cc}\frac{2\left|\lambda \right|x}{{\left[2\right]}_q{\left[3\right]}_q},\hfill & \left|1-\sigma \right|\le ϝ,\hfill \\ & \\ \frac{16\left(1-\sigma \right){\left|{\left[\lambda \right]}_q\right|}^3{x}^3}{{\left[2\right]}_q\left(\left(2\left(\left(2{\left[3\right]}_{q}2-3{\left[2\right]}_q\right){\left[\lambda \right]}_q^2-2{\left[2\right]}_q{\left[\lambda \right]}_{q^2}\right)\right)x^2+{\left[2\right]}_q{\left[\lambda \right]}_{q^2}\right)},\hfill & \left|1-\sigma \right|\ge ϝ,\hfill \end{array}\right.$$

where

$$ϝ=\frac{\left(2\left(\left(2{\left[3\right]}_{q}2-3{\left[2\right]}_q\right){\left[\lambda \right]}_q^2-2{\left[2\right]}_q{\left[\lambda \right]}_{q^2}\right)\right)x^2+{\left[2\right]}_q{\left[\lambda \right]}_{q^2}}{16{\left[2\right]}_q{\left[3\right]}_q{\left[\lambda \right]}_q^2{x}^2}.$$

Proof. 

From (50) and (51),

$$\begin{array}{cc}\hfill a_3-\sigma a_2^2& =\left(1-\sigma \right)\frac{{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^3}{2{\left[2\right]}_q\left(q^2{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2-{\left[2\right]}_q{C}_2^{\left(\lambda \right)}(x;q)\right)}\left(c_2+d_2\right)\hfill \\ \hfill & +\frac{C_1^{\left(\lambda \right)}(x;q)}{2{\left[2\right]}_q{\left[3\right]}_q}\left(c_2-d_2\right)\hfill \\ \hfill & =C_1^{\left(\lambda \right)}(x;q)\left[\left[\mathcal{K}\left(\sigma \right)+\frac{1}{2{\left[2\right]}_q{\left[3\right]}_q}\right]c_2+\left[\mathcal{K}\left(\sigma \right)-\frac{1}{2{\left[2\right]}_q{\left[3\right]}_q}\right]d_2\right],\hfill \end{array}$$

where

$$\mathcal{K}\left(\sigma \right)=\frac{\left(1-\sigma \right){\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2}{2{\left[2\right]}_q\left(q^2{\left[C_1^{\left(\lambda \right)}(x;q)\right]}^2-{\left[2\right]}_q{C}_2^{\left(\lambda \right)}(x;q)\right)},$$

$$\left|1-\sigma \right|\le \frac{\left(4q^2{\left[\lambda \right]}_q^2{x}^2-{\left[2\right]}_q{C}_2^{\left(\lambda \right)}(x;q)\right)}{4{\left[3\right]}_q{\left[\lambda \right]}_q^2{x}^{}}$$

Then, in view of (5), we conclude that

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{\left|C_1^{\left(\lambda \right)}(x;q)\right|}{{\left[2\right]}_q{\left[3\right]}_q},\hfill & \left|\mathcal{K}\left(\sigma \right)\right|\le \frac{1}{2{\left[2\right]}_q{\left[3\right]}_q},\hfill \\ & \\ \frac{1}{{\left[2\right]}_q}\left|C_1^{\left(\lambda \right)}(x;q)\right|\left|\mathcal{K}\left(\sigma \right)\right|,\hfill & \left|\mathcal{K}\left(\sigma \right)\right|\ge \frac{1}{2{\left[2\right]}_q{\left[3\right]}_q}.\hfill \end{array}\right.$$

This completes the proof of the last theorem. □

Corollary 4. 

Let $f\in \Sigma$ given by (1) belong to the class ${\mathfrak{C}}_{\Sigma }(x,\alpha ;1)$. Then,

$$|a_2|\le \frac{\lambda x\sqrt{2x}}{\sqrt{1-2x^2}},$$

$$|a_3|\le {\lambda }^2{x}^2+\frac{\lambda x}{3}.$$

and

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{2\left|{\left[\lambda \right]}_q\right|x}{{\left[2\right]}_q{\left[3\right]}_q},\hfill & |\sigma -1|\le \left|\frac{1-2\lambda x^2}{24\lambda x^2}\right|,\hfill \\ \frac{1}{{\left[2\right]}_q}\left|C_1^{\left(\lambda \right)}(x;q)\right|\left|\mathcal{K}\left(\sigma \right)\right|,\hfill & |\sigma -1|\ge \left|\frac{1-2\lambda x^2}{24\lambda x^2}\right|.\hfill \end{array}\right.$$

6. Conclusions

In the current study, we introduced and examined the coefficient issues related to each of the three new subclasses of the class of bi-univalent functions in the open unit disk $\mathbb{U}$: ${\mathfrak{B}}_{\Sigma }(x,\alpha ;q)$, ${\mathcal{S}}_{\Sigma }^{*}(x,\alpha ;q)$, and ${\mathfrak{C}}_{\Sigma }(x,\alpha ;q)$. These bi-univalent function classes are described, accordingly, in Definitions 1 to 3. We calculated the estimates of the Fekete–Szegö functional problems and the Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ for functions in each of these three bi-univalent function classes. Several more fresh outcomes are revealed to follow following specializing the parameters involved in our main results. Obtaining estimates on the bound of $\left|a_n\right|$ for $n\ge 4;n\in \mathbb{N}$ for the classes that have been introduced here is still a problem.