Applications of Gegenbauer Polynomials for Subfamilies of Bi-Univalent Functions Involving a Borel Distribution-Type Mittag-Leffler Function

Abstract

In this research, a novel linear operator involving the Borel distribution and Mittag-Leffler functions is introduced using Hadamard products or convolutions. This operator is utilized to develop new subfamilies of bi-univalent functions via the principle of subordination with Gegenbauer orthogonal polynomials. The investigation also focuses on the estimation of the coefficients $|a_{\ell }|(\ell =2,3)$ and the Fekete–Szegö inequality for functions belonging to these subfamilies of bi-univalent functions. Several corollaries and implications of the findings are discussed. Overall, this study presents a new approach for constructing bi-univalent functions and provides valuable insights for further research in this area.

1. Introduction

The distribution of probabilities over the values of a random variable is a critical concept in statistics and probability. This concept is widely used to describe and model various real-life phenomena [1]. To highlight the importance of specific distributions and their associated random experiments, these distributions are assigned specific names. For instance, when considering the number of repetitions required for success in a random experiment with two outcomes, the geometric distribution is employed. The geometric distribution’s name is derived from the relationship between the distribution and the geometric series.

In probability theory and statistics, two vital concepts are the symmetry and distributions of random variables. Symmetry refers to an object or system’s property that remains the same under certain transformations. In probability theory, symmetry is frequently used to describe a random variable’s distribution, which is a function that maps the outcomes of a random experiment to real numbers.

Orthogonal polynomials are a fundamental concept in mathematical analysis and have been studied extensively since their inception in the 19th century [2]. These polynomials form a sequence of functions that are orthogonal concerning a specific weight function on a given interval. They have been used in many areas of mathematics, including number theory, approximation theory, and the theory of differential equations.

One of the most important classes of orthogonal polynomials is the classical orthogonal polynomials, which include the Gegenbauer, Legendre, Hermite, and Laguerre polynomials. These polynomials have been extensively studied and form the basis of many mathematical techniques and methods.

Let ${\mathsf{P}}_s$ and ${\mathsf{P}}_j$ be polynomials of order s and j. Subsequently, ${\mathsf{P}}_s$ and ${\mathsf{P}}_j$ are orthogonal polynomials over $[\mathsf{a},\mathsf{b}]$, if there is a non-negative function $\nu (u)$ and

$$\langle {\mathsf{P}}_j,{\mathsf{P}}_s\rangle ={\int }_{\mathsf{a}}^{\mathsf{b}}{\mathsf{P}}_j(u){\mathsf{P}}_s(u)\nu (u)du=0,(s\ne j),$$

all finite-order polynomials ${\mathsf{P}}_j(u)$ have an integral that is clearly defined (see [3,4]).

Gegenbauer polynomials fall within the category of orthogonal polynomials. According to [4,5,6], where conventional algebraic formulations are employed, a symbolic relationship exists between the generating function of Gegenbauer polynomials and the integral representation of usually real functions $T_R$, which consequently led to the emergence of certain significant inequalities from the realm of Gegenbauer polynomials.

In 2021, Amourah et al. [7,8] employed the Gegenbauer function ${\mathfrak{G}}^{(\zeta )}(\gimel ,\eta )$ for nonzero real constant $\zeta$ given by

$${\mathfrak{G}}^{(\zeta )}(\gimel ,\eta )=\frac{1}{{\left(1-2\gimel \eta +{\eta }^2\right)}^{\zeta }},\gimel \in [-1,1],$$

(1)

for a fixed ℷ and analytic function ${\mathfrak{G}}^{(\zeta )}$, we can write

$${\mathfrak{G}}^{(\zeta )}(\gimel ,\eta )=\sum _{\ell =0}^{\infty }{C}_{\ell }^{\zeta }(\gimel ){\eta }^{\ell },$$

where $C_{\ell }^{\zeta }(\gimel )$ is the Gegenbauer polynomial.

In addition, the recurrence relations that characterize Gegenbauer polynomials are as follows:

$$C_{\ell }^{\zeta }(\gimel )=\frac{1}{\ell }\left[2\gimel \left(\ell +\zeta -1\right)C_{\ell -1}^{\zeta }(\gimel )-\left(\ell +2\zeta -2\right)C_{\ell -2}^{\zeta }(\gimel )\right].$$

(2)

The first few terms of Gegenbauer polynomials are given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathcal{C}}_0^{(\zeta )}(\gimel )& =1,{\mathcal{C}}_1^{(\zeta )}(\gimel )=2\zeta \gimel ,{\mathcal{C}}_2^{(\zeta )}(\gimel )=2\left(\zeta +{\zeta }^2\right){\gimel }^2-\zeta .\hfill \end{array}\end{array}$$

(3)

Note that, for $\zeta =1$ or $\zeta =\frac{1}{2}$, we get the Chebyshev Polynomials ${\mathcal{C}}_{\ell }^{(1)}(\gimel )$ and Legendre Polynomials ${\mathcal{C}}_{\ell }^{(\frac{1}{2})}(\gimel )$, respectively.

2. Preliminaries

Let $\mathfrak{A}$ represent the family of analytic functions f in ∇, where

$$f(\eta )=\eta +\sum _{\ell =2}^{\infty }{a}_{\ell }{\eta }^{\ell },(\eta \in \nabla )$$

(4)

and

$$\nabla =\{\eta \in \mathbb{C}:\left|\eta \right|<1\},$$

with the normalized condition $f^{\prime }(0)-1=0=f(0)$. We also represent by $\mathcal{S}$ the subfamily of $\mathfrak{A}$, which is also univalent in ∇.

The use of differential subordination in the study of analytical functions can greatly enhance the field of geometric function theory. The initial problem of differential subordination was first presented by Miller and Mocanu [9], and subsequent developments in the field can be found in [10] and their book [11], which serves as a comprehensive reference for the subject.

Suppose that $f^{-1}$ is the inverse of the function $f\in \mathfrak{A}$, which is expressed by

$$\eta =f^{-1}(f(\eta ))\mathrm{and}\varpi =f(f^{-1}(\varpi ))\left(r_0(f)\ge \frac{1}{4};\left|\varpi \right|<r_0(f)\right),$$

where

$$\hslash (\varpi )=f^{-1}(\varpi )=\varpi -a_2{\varpi }^2+(-a_3+2a_2^2){\varpi }^3-(a_4+5a_2^3-5a_3{a}_2){\varpi }^4+\cdots .$$

(5)

If both $f(\eta )$ and $f^{-1}(\varpi )$ are univalent in ∇, then the function is called bi-univalent in ∇. The family of all univalent functions in ∇ is denoted by $\Sigma$. The instances of functions in the family $\Sigma$ are

$$\frac{\eta }{1-\eta },log\left(\frac{1}{1-\eta }\right).$$

Elementary distributions are a class of generalized functions that have been extensively studied in mathematical analysis. These functions are used to represent a wide range of mathematical objects, including point masses, impulse functions, and derivatives of distributions. Theoretically, in Geometric Function Theory, the elementary distributions, including the Poisson, Pascal, Logarithmic, and Binomial, have been investigated (see [12,13,14,15,16]).

Recently, Wanas and Khuttar [12] developed the power series whose coefficients represent probabilities of the Borel distribution

$$\mathfrak{M}(\mu ,\eta )=\eta +\sum _{\ell =2}^{\infty }\frac{{(\mu (\ell -1))}^{\ell -2}{e}^{-\mu (\ell -1)}}{(\ell -1)!}{\eta }^{\ell },(0<\mu \le 1;\eta \in \nabla ).$$

(6)

The above-mentioned series converges with the domain of convergence of the entire complex plane using a well-known ratio test. Hence, this study takes into account the subsequent normalization of the Mittag-Leffler function (see [17,18,19,20,21]):

$${\mathfrak{E}}_{\alpha ,\vartheta }(\eta )=\eta +\sum _{\ell =2}^{\infty }\frac{\Gamma (\vartheta )}{\Gamma (\alpha (\ell -1)+\vartheta )}{\eta }^{\ell },(\eta \in \nabla ),$$

(7)

where $\alpha ,\vartheta \in \mathbb{C};\vartheta \ne 0,-1,-2,\cdots$; and $\Re e(\alpha )>0,\Re e(\vartheta )>0$.

Utilizing the convolution (or the Hadamard products) notion (∗) of two analytic-functions, we have

$$\begin{array}{cc}\hfill {\mathfrak{N}}_{\alpha ,\vartheta }^{\mu }(\eta )& ={\mathfrak{E}}_{\alpha ,\vartheta }(\eta )\ast \mathfrak{M}(\mu ,\eta )\hfill \\ & =\eta +\sum _{\ell =2}^{\infty }\frac{\Gamma (\vartheta ){(\mu (\ell -1))}^{\ell -2}{e}^{-\mu (\ell -1)}}{\Gamma (\alpha (\ell -1)+\vartheta )(\ell -1)!}{\eta }^{\ell },\hfill \end{array}$$

(8)

where $0<\mu \le 1$, $\alpha ,\vartheta \in \mathbb{C}; \vartheta \ne 0,-1,-2,\cdots$; and $\Re e(\alpha )>0,\Re e(\vartheta )>0$. Now, we define a linear operator ${\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }(\eta ):\mathfrak{A}\to \mathfrak{A}$ as below:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }(\eta )& ={\mathfrak{N}}_{\alpha ,\vartheta }^{\mu }(\eta )\ast f(\eta )=\eta +\sum _{\ell =2}^{\infty }\frac{\Gamma (\vartheta ){(\mu (\ell -1))}^{\ell -2}{e}^{-\mu (\ell -1)}}{\Gamma (\alpha (\ell -1)+\vartheta )(\ell -1)!}{a}_{\ell }{\eta }^{\ell },\hfill \end{array}\end{array}$$

(9)

where $0<\mu \le 1$, $\alpha ,\vartheta \in \mathbb{C};\vartheta \ne 0,-1,-2,\cdots$ and $\Re e(\alpha )>0,\Re e(\vartheta )>0$.

In the recent past, there has been a growing interest among researchers to investigate specific subsets of bi-univalent functions associated with orthogonal polynomials. Through their work, they have been able to derive estimates for the initial coefficients of these functions. Nonetheless, despite the research effort ([22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]) the question of obtaining precise bounds for the coefficients $|a_{\ell }|,(\ell =3,4,5,\cdots )$ remains unanswered.

Numerous scholars have conducted research on particular subclasses of bi-univalent functions, utilizing a variety of probability distributions, including the Pascal, Poisson, and Borel distributions (for instance, see [41,42,43,44,45]). The primary aim of this study is to investigate the features of bi-univalent functions concerning Gegenbauer polynomials. It is essential to bear the following definitions in mind while delving into this research.

3. The Class ${\mathcal{B}}_{\Sigma }^{\mathbf{\delta }}(\mathbf{\alpha },\mathbf{\vartheta },\mathbf{\mu },{\mathfrak{G}}^{(\mathbf{\zeta })}(\gimel ,\mathbf{\eta }))$

This part defines and studies a novel subfamily of bi-univalent functions using the principle of subordination and the linear operator ${\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }(\eta ):\mathfrak{A}\to \mathfrak{A}$ given by (9) and the ordinary Gegenbauer polynomials.

Definition 1. 

A bi-univalent function f of the form (4) belongs to the family ${\mathcal{B}}_{\Sigma }^{\delta }(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$ if the below subordination conditions hold:

$$\begin{array}{cc}\hfill (1-\delta )\frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )}{\eta }+\delta {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )\right)}^{\prime }& \prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\eta )\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill (1-\delta )\frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )}{\varpi }+\delta {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )\right)}^{\prime }& \prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\varpi ),\hfill \end{array}$$

(11)

where $\zeta >0$, $\delta \ge 0$, $\gimel \in \left(\frac{1}{2},1\right]$, and the function $\hslash =f^{-1}$ is given by (5).

According to the parameter $\mu$, we deduce subfamilies of $\Sigma$ as follows:

Example 1. 

A bi-univalent function f of the form (4) belongs to the family ${\mathcal{B}}_{\Sigma }^1(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$ if the below subordination conditions hold:

$$\begin{array}{cc}\hfill {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )\right)}^{\prime }& \prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ),\hfill \\ \hfill & \\ \hfill {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )\right)}^{\prime }& \prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\varpi ),\hfill \end{array}$$

where $\zeta >0$, $\gimel \in \left(\frac{1}{2},1\right]$, and the function $\hslash =f^{-1}$ has been mentioned previously in (5).

Example 2. 

A bi-univalent function f of the form (4) belongs to the family ${\mathcal{B}}_{\Sigma }^0(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$ if the below subordination conditions hold:

$$\begin{array}{cc}\hfill \frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )}{\eta }& \prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ),\hfill \\ \hfill \frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )}{\varpi }& \prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\varpi ),\hfill \end{array}$$

where $\zeta >0$, $\gimel \in \left(\frac{1}{2},1\right]$, and the function $\hslash =f^{-1}$ has been mentioned previously in (5).

4. Estimates of the Family ${\mathcal{B}}_{\Sigma }^{\mathbf{\delta }}(\mathbf{\alpha },\mathbf{\vartheta },\mathbf{\mu },{\mathfrak{G}}^{(\mathbf{\zeta })}(\gimel ,\mathbf{\eta }))$

This part determines coefficient estimates for the family ${\mathcal{B}}_{\Sigma }^{\delta }(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$ given in Definition 1.

Theorem 1. 

Let f in (4) belong to family ${\mathcal{B}}_{\Sigma }^{\delta }(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$; then,

$$|a_2|\le \frac{2|\zeta |\gimel e^{\mu }\Gamma (\alpha +\vartheta )\sqrt{2\zeta \gimel \Gamma (2\alpha +\vartheta )}}{\sqrt{\Gamma (\vartheta )\left(\begin{array}{c}\left(2(1+2\delta )\mu {\left(\Gamma (\alpha +\vartheta )\right)}^2{\zeta }^2-2\zeta \Gamma (\vartheta )\Gamma (2\alpha +\vartheta ){(1+\delta )}^2(1+\zeta )\right){\gimel }^2\\ +2\Gamma (\vartheta )\Gamma (2\alpha +\vartheta ){(1+\delta )}^2\end{array}\right)}},$$

and

$$|a_3|\le \frac{1}{2}{\left(\frac{2\zeta \gimel e^{\mu }\Gamma (\alpha +\vartheta )}{(1+\delta )\Gamma (\vartheta )}\right)}^2+\frac{4\zeta \gimel e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}.$$

Proof. 

Let $f\in {\mathcal{B}}_{\Sigma }^{\delta }(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$. From Definition 1, for some analytical $\omega ,\vartheta$ with initial conditions $\omega (0)=\vartheta (0)=0$ and $|\omega (\eta )|<1$, $|\vartheta (\phi )|<1$ ($\eta ,\varpi \in \nabla$), then

$$(1-\delta )\frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )}{\eta }+\delta {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )\right)}^{\prime }\prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\omega (\eta )),$$

(12)

$$(1-\delta )\frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )}{\varpi }+\delta {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )\right)}^{\prime }\prec {\mathfrak{G}}^{(\zeta )}(\gimel ,\vartheta (\varpi )).$$

(13)

From the Equations (12) and (13), we obtain that

$$(1-\delta )\frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )}{\eta }+\delta {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }f(\eta )\right)}^{\prime }=1+C_1^{(\zeta )}(\gimel )c_1\eta +\left[C_1^{(\zeta )}(\gimel )c_2+C_2^{(\zeta )}(\gimel )c_1^2\right]{\eta }^2+\cdots ,$$

(14)

and

$$(1-\delta )\frac{{\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )}{\varpi }+\delta {\left({\mathfrak{L}}_{\alpha ,\vartheta }^{\mu }\hslash (\varpi )\right)}^{\prime }=1+C_1^{(\zeta )}(\gimel )d_1\varpi +\left[C_1^{(\zeta )}(\gimel )d_2+C_2^{(\zeta )}(\gimel )d_1^2\right]){\varpi }^2+\cdots ,$$

(15)

where $\omega (\eta )$ and $v(\varpi )$ have the form

$$\left|\omega (\eta )\right|=\left|c_1\eta +c_2{\eta }^2+c_3{\eta }^3+\cdots \right|<1,(\eta \in \nabla )$$

and

$$\left|v(\varpi )\right|=\left|d_1\varpi +d_2{\varpi }^2+d_3{\varpi }^3+\cdots \right|<1,(\varpi \in \nabla ),$$

and then, for all $(j\in \mathbb{N})$, yields

$$|c_j|\le 1\mathrm{and}|d_j|\le 1.$$

(16)

The equivalent coefficients in (14) and (15) are compared; then,

$$\frac{(1+\delta )\Gamma (\vartheta )}{e^{\mu }\Gamma (\alpha +\vartheta )}{a}_2=C_1^{(\zeta )}(\gimel )c_1,$$

(17)

$$\frac{(1+2\delta )\mu \Gamma (\vartheta )}{2e^{2\mu }\Gamma (2\alpha +\vartheta )}{a}_3=C_1^{(\zeta )}(\gimel )c_2+C_2^{(\zeta )}(\gimel )c_1^2,$$

(18)

$$-\frac{(1+\delta )\Gamma (\vartheta )}{e^{\mu }\Gamma (\alpha +\vartheta )}{a}_2=C_1^{(\zeta )}(\gimel )d_1,$$

(19)

and

$$\frac{(1+2\delta )\mu \Gamma (\vartheta )}{2e^{2\mu }\Gamma (2\alpha +\vartheta )}\left(2a_2^2-a_3\right)=C_1^{(\zeta )}(\gimel )d_2+C_2^{(\zeta )}(\gimel )d_1^2.$$

(20)

It follows from (17) and (19) that

$$c_1=-d_1$$

(21)

and

$$2{\left(\frac{(1+\delta )\Gamma (\vartheta )}{e^{\mu }\Gamma (\alpha +\vartheta )}\right)}^2{a}_2^2={\left[C_1^{(\zeta )}(\gimel )\right]}^2\left(c_1^2+d_1^2\right).$$

(22)

By adding (18) to (20), we get

$$\frac{(1+2\delta )\mu \Gamma (\vartheta )}{e^{2\mu }\Gamma (2\alpha +\vartheta )}{a}_2^2=C_1^{(\zeta )}(\gimel )(c_2+d_2)+C_2^{(\zeta )}(\gimel )(c_1^2+d_1^2).$$

(23)

Substituting the estimate of $\left(c_1^2+d_1^2\right)$ from (22) in the Equation (23), we conclude that

$$\left(\frac{(1+2\delta )\mu }{\Gamma (2\alpha +\vartheta )}-\frac{2\Gamma (\vartheta ){(1+\delta )}^2}{{\left(\Gamma (\alpha +\vartheta )\right)}^2}\frac{C_2^{(\zeta )}(\gimel )}{{\left[C_1^{(\zeta )}(\gimel )\right]}^2}\right)\frac{\Gamma (\vartheta )}{e^{2\mu }}{a}_2^2=C_1^{(\zeta )}(\gimel )(c_2+d_2).$$

(24)

By using (3) along to (24), we find that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |a_2|& \le \frac{2|\zeta |\gimel e^{\mu }\Gamma (\alpha +\vartheta )\sqrt{2\zeta \gimel \Gamma (2\alpha +\vartheta )}}{\sqrt{\Gamma (\vartheta )\left(2(1+2\delta )\mu {\left(\Gamma (\alpha +\vartheta )\right)}^2{\zeta }^2{\gimel }^2\right)-\Gamma (\vartheta )\Gamma (2\alpha +\vartheta ){(1+\delta )}^2\left(2\zeta (1+\zeta ){\gimel }^2-\gimel \right)}}.\hfill \end{array}\end{array}$$

(25)

Moreover, if we subtract (20) from (18), we obtain

$$\frac{(1+2\delta )\mu \Gamma (\vartheta )}{e^{2\mu }\Gamma (2\alpha +\vartheta )}\left(a_3-a_2^2\right)=C_1^{(\zeta )}(\gimel )(c_2-d_2)+C_2^{(\zeta )}(\gimel )(c_1^2-d_1^2).$$

(26)

In view of (22) and (26) becomes

$$a_3=\frac{1}{2}{\left(\frac{e^{\mu }\Gamma (\alpha +\vartheta )}{(1+\delta )\Gamma (\vartheta )}\right)}^2{\left[C_1^{(\zeta )}(\gimel )\right]}^2\left(c_1^2+d_1^2\right)+\frac{e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}{C}_1^{(\zeta )}(\gimel )(c_2-d_2).$$

By using the Equation (24), we find that

$$|a_3|\le \frac{1}{2}{\left(\frac{2\zeta \gimel e^{\mu }\Gamma (\alpha +\vartheta )}{(1+\delta )\Gamma (\vartheta )}\right)}^2+\frac{4\zeta \gimel e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}.$$

Here, the demonstration completes.

Inspired by the outcome of Zaprawa’s investigation [46], this research demonstrates the Fekete–Szegö problem for family ${\mathcal{B}}_{\Sigma }^{\delta }(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$ by adopting the values of $a_2^2$ and $a_3$. □

Theorem 2. 

Let f in (4) belong to family ${\mathcal{B}}_{\Sigma }^{\delta }(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$; then,

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{4e^{2\mu }\Gamma (2\alpha +\vartheta )|\zeta |\gimel }{(1+2\delta )\mu \Gamma (\vartheta )},\hfill & \left|\sigma -1\right|\le v(\gimel ),\hfill \\ \\ \frac{{8|\zeta |}^3{\gimel }^3{e}^{2\mu }\Gamma (2\alpha +\vartheta ){\left(\Gamma (\alpha +\vartheta )\right)}^2|1-\sigma |}{\left|\Gamma (\vartheta )\left(4{\zeta }^2{\gimel }^2{\left(\Gamma (\alpha +\vartheta )\right)}^2-\Gamma (2\alpha +\vartheta )(2\left(\zeta +{\zeta }^2\right){\gimel }^2-\zeta )\right)\right|},\hfill & \left|\sigma -1\right|\ge v(\gimel ),\hfill \end{array}\right.$$

where

$$v(\gimel )=\left|\left(1-\frac{\Gamma (2\alpha +\vartheta )\left(2\left(\zeta +{\zeta }^2\right){\gimel }^2-\zeta \right)}{4{\zeta }^2{\gimel }^2{\left(\Gamma (\alpha +\vartheta )\right)}^2}\right)\frac{1}{(1+2\delta )\mu }\right|.$$

Proof. 

Let $f\in {\mathcal{B}}_{\Sigma }^{\delta }(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$. Then, from the Equations (24) and (26), we have

$$\left(\frac{(1+2\delta )\mu }{\Gamma (2\alpha +\vartheta )}-\frac{2\Gamma (\vartheta ){(1+\delta )}^2}{{\left(\Gamma (\alpha +\vartheta )\right)}^2}\frac{C_2^{(\zeta )}(\gimel )}{{\left[C_1^{(\zeta )}(\gimel )\right]}^2}\right)\frac{\Gamma (\vartheta )}{e^{2\mu }}{a}_2^2=C_1^{(\zeta )}(\gimel )(c_2+d_2),$$

(27)

and

$$a_3=a_2^2+\frac{e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}{C}_1^{(\zeta )}(\gimel )(c_2-d_2).$$

Rearranging this, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_3-\sigma a_2^2& =\frac{(1-\sigma )e^{2\mu }\Gamma (2\alpha +\vartheta ){\left(\Gamma (\alpha +\vartheta )\right)}^2{\left[C_1^{(\zeta )}(\gimel )\right]}^3}{\Gamma (\vartheta )\left({\left(\Gamma (\alpha +\vartheta )\right)}^2{\left[C_1^{(\zeta )}(\gimel )\right]}^2-\Gamma (2\alpha +\vartheta )C_2^{(\zeta )}(\gimel )\right)}(c_2+d_2)\hfill \\ \hfill & +\frac{e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}{C}_1^{(\zeta )}(\gimel )(c_2-d_2)\hfill \\ \hfill & =C_1^{(\zeta )}(\gimel )\left[\mathfrak{h}(\sigma )+\frac{e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}\right]c_2+C_1^{(\zeta )}(\gimel )\left[\mathfrak{h}(\sigma )-\frac{e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}\right]d_2\hfill \end{array}\end{array}$$

where

$$\mathfrak{h}(\sigma )=\frac{(1-\sigma )e^{2\mu }\Gamma (2\alpha +\vartheta ){\left(\Gamma (\alpha +\vartheta )\right)}^2{\left[C_1^{(\zeta )}(\gimel )\right]}^2}{\Gamma (\vartheta )\left({\left(\Gamma (\alpha +\vartheta )\right)}^2{\left[C_1^{(\zeta )}(\gimel )\right]}^2-\Gamma (2\alpha +\vartheta )C_2^{(\zeta )}(\gimel )\right)}.$$

Then, in view of (3), we conclude that

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{2e^{2\mu }\Gamma (2\alpha +\vartheta )\left|C_1^{(\zeta )}(\gimel )\right|}{(1+2\delta )\mu \Gamma (\vartheta )},\hfill & \left|\mathfrak{h}(\sigma )\right|\le \frac{e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )},\hfill \\ \\ 2\left|C_1^{(\zeta )}(\gimel )\right|\left|\mathfrak{h}(\sigma )\right|,\hfill & \left|\mathfrak{h}(\sigma )\right|\ge \frac{e^{2\mu }\Gamma (2\alpha +\vartheta )}{(1+2\delta )\mu \Gamma (\vartheta )}.\hfill \end{array}\right.$$

□

5. Corollaries and Consequences

Theorems 1 and 2 generate the following result, which roughly corresponds to Examples 1 and 2.

Corollary 1. 

Let f in (4) belong to family ${\mathcal{B}}_{\Sigma }^0(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$; then,

$$\begin{array}{cc}\hfill |a_2|& \le \frac{2|\zeta |\gimel e^{\mu }\Gamma (\alpha +\vartheta )\sqrt{2\zeta \gimel \Gamma (2\alpha +\vartheta )}}{\sqrt{\Gamma (\vartheta )\left(2\mu {\left(\Gamma (\alpha +\vartheta )\right)}^2{\zeta }^2{\gimel }^2\right)-\Gamma (\vartheta )\Gamma (2\alpha +\vartheta )\left(2\zeta (1+\zeta ){\gimel }^2-\gimel \right)}},\hfill \\ \hfill |a_3|& \le 2{\left(\frac{\zeta \gimel e^{\mu }\Gamma (\alpha +\vartheta )}{\Gamma (\vartheta )}\right)}^2+\frac{4\zeta \gimel e^{2\mu }\Gamma (2\alpha +\vartheta )}{\mu \Gamma (\vartheta )},\hfill \end{array}$$

and

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{4e^{2\mu }\Gamma (2\alpha +\vartheta )|\zeta |\gimel }{\mu \Gamma (\vartheta )},\hfill & \left|\sigma -1\right|\le v(\gimel ),\hfill \\ \\ \frac{{8|\zeta |}^3{\gimel }^3{e}^{2\mu }\Gamma (2\alpha +\vartheta ){\left(\Gamma (\alpha +\vartheta )\right)}^2|1-\sigma |}{\left|\Gamma (\vartheta )\left(4{\zeta }^2{\gimel }^2{\left(\Gamma (\alpha +\vartheta )\right)}^2-\Gamma (2\alpha +\vartheta )(2\left(\zeta +{\zeta }^2\right){\gimel }^2-\zeta )\right)\right|},\hfill & \left|\sigma -1\right|\ge v(\gimel ),\hfill \end{array}\right.$$

where

$$v(\gimel )=\left|\frac{1}{\mu }\left(1-\frac{\Gamma (2\alpha +\vartheta )\left(2\left(\zeta +{\zeta }^2\right){\gimel }^2-\zeta \right)}{4{\zeta }^2{\gimel }^2{\left(\Gamma (\alpha +\vartheta )\right)}^2}\right)\right|.$$

Corollary 2. 

Let f in (4) belong to family ${\mathcal{B}}_{\Sigma }^1(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$; then,

$$\begin{array}{cc}\hfill |a_2|& \le \frac{2|\zeta |\gimel e^{\mu }\Gamma (\alpha +\vartheta )\sqrt{2\zeta \gimel \Gamma (2\alpha +\vartheta )}}{\sqrt{\Gamma (\vartheta )\left(6\mu {\left(\Gamma (\alpha +\vartheta )\right)}^2{\zeta }^2{\gimel }^2\right)-4\Gamma (\vartheta )\Gamma (2\alpha +\vartheta )\left(2\zeta (1+\zeta ){\gimel }^2-\gimel \right)}},\hfill \\ \hfill |a_3|& \le \frac{1}{2}{\left(\frac{\zeta \gimel e^{\mu }\Gamma (\alpha +\vartheta )}{\Gamma (\vartheta )}\right)}^2+\frac{4\zeta \gimel e^{2\mu }\Gamma (2\alpha +\vartheta )}{3\mu \Gamma (\vartheta )},\hfill \end{array}$$

and

$$|a_3-\sigma a_2^2|\le \left\{\begin{array}{cc}\frac{4e^{2\mu }\Gamma (2\alpha +\vartheta )|\zeta |\gimel }{3\mu \Gamma (\vartheta )},\hfill & \left|\sigma -1\right|\le v(\gimel ),\hfill \\ \\ \frac{{8|\zeta |}^3{\gimel }^3{e}^{2\mu }\Gamma (2\alpha +\vartheta ){\left(\Gamma (\alpha +\vartheta )\right)}^2|1-\sigma |}{\left|\Gamma (\vartheta )\left(4{\zeta }^2{\gimel }^2{\left(\Gamma (\alpha +\vartheta )\right)}^2-\Gamma (2\alpha +\vartheta )(2\left(\zeta +{\zeta }^2\right){\gimel }^2-\zeta )\right)\right|},\hfill & \left|\sigma -1\right|\ge v(\gimel ),\hfill \end{array}\right.$$

where

$$v(\gimel )=\left|\frac{1}{3\mu }\left(1-\frac{\Gamma (2\alpha +\vartheta )\left(2\left(\zeta +{\zeta }^2\right){\gimel }^2-\zeta \right)}{4{\zeta }^2{\gimel }^2{\left(\Gamma (\alpha +\vartheta )\right)}^2}\right)\right|.$$

6. Conclusions

In this study, we introduced a new category of bi-univalent functions that comprises three novel subfamilies: ${\mathcal{B}}^{\delta }\Sigma (\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$, ${\mathcal{B}}^1\Sigma (\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}(\gimel ,\eta ))$, and ${\mathcal{B}}_{\Sigma }^0(\alpha ,\vartheta ,\mu ,{\mathfrak{G}}^{(\zeta )}$$(\gimel ,\eta )).$ We then addressed the coefficient complexities associated with these new subfamilies by defining them via Definition 1. We also derived the upper bounds of the Fekete–Szegö inequality and initial coefficient estimates $|a_2|$ and $|a_3|$ for functions in each of these three subfamilies. In addition to discussing the parameters utilized in this study, we presented several discoveries, although the results obtained are not precise and are open to others to demonstrate their sharpness. Our aim in this paper is to illustrate significant outcomes that may inspire further research into this concept for analytic bi-univalent functions and symmetric q-calculus. Furthermore, the symmetry q-sine domain and symmetry q-cosine domain could be utilized in place of the given domain.