Subfamilies of Bi-Univalent Functions Associated with the Imaginary Error Function and Subordinate to Jacobi Polynomials

Abstract

Numerous researchers have extensively studied various subfamilies of the bi-univalent function family utilizing special functions. In this paper, we introduce and investigate a new subfamily of bi-univalent functions, which is defined on the symmetric domain. This subfamily is connected to the Jacobi polynomial through the imaginary error function. We derive the initial coefficients of the Maclaurin series for functions in this subfamily, and analyze the Fekete–Szegő inequality for these functions. Additionally, by specializing the parameters in our main results, we deduce several new and significant findings.

1. Introduction and Preliminaries

Ordinary differential equations that satisfy model constraints are often solved using orthogonal polynomials [1]. These polynomials are fundamental in physics and engineering, and hold significant importance in modern mathematics, particularly in approximation theory. Orthogonal polynomials are applied in various fields, including quantum physics, probability theory, interpolation, differential equation theory, and mathematical statistics [2]. Additionally, they are used to model and analyze complex systems and datasets in signal processing, image processing, and data analysis (see [3,4,5]).

The pair of polynomials, ${\mathbb{J}}_{ϵ}$ and ${\mathbb{J}}_{\epsilon }$, of orders $ϵ$ and $\epsilon$, respectively, are orthogonal if

$$\langle {\mathbb{J}}_{ϵ},{\mathbb{J}}_{\epsilon }\rangle ={\int }_{{\sigma }_1}^{{\sigma }_2}{\mathbb{J}}_{ϵ}\left(y\right){\mathbb{J}}_{\epsilon }\left(y\right)r\left(y\right)dy=0,\mathrm{for}ϵ\ne \epsilon .$$

(1)

The integral of all finite-order polynomials ${\mathbb{J}}_{ϵ}\left(y\right)$ is properly defined, as $r\left(y\right)$ is a non-negative function in the interval $({\sigma }_1,{\sigma }_2)$ (see [6]).

Many families of orthogonal polynomials, such as Laguerre, Legendre, Hermite, Chebyshev, and so on, are well-known. These polynomials possess many useful properties and applications. Each family is characterized by its weight function and interval.

The generating function of Jacobi polynomials is defined by

$$J_{\nu }(t,z)=2^{\alpha +\beta }{R}^{-1}{\left(1-t+R\right)}^{-\alpha }{\left(1+t+R\right)}^{-\beta },$$

where $R:=R(t,z)={(1-2zt+t^2)}^{0.5}$, $\alpha$, and $\beta >-1$, $t\in [-1,1]$, $\nu ,\nu +\alpha ,\nu +\beta$ are non-negative integers, and z in the the open unit disk $\mathbb{D}=\{z\in \mathbb{C}:\left|z\right|<1\}$, (see [7]).

For a fixed t, the function $J_{\nu }(t,z)$ is analytic in $\mathbb{D}$, allowing it to be represented by a Taylor series expansion as follows:

$$J_{\nu }(t,z)=\sum _{\nu =0}^{\infty }{P}_{\nu }^{(\alpha ,\beta )}\left(t\right)z^{\nu },$$

(2)

where $P_{\nu }^{(\alpha ,\beta )}\left(t\right)$ is Jacobi polynomial of degree $\nu$.

The Jacobi polynomial $P_{\nu }^{(\alpha ,\beta )}\left(t\right)$ satisfies the second-order linear homogeneous differential equation,

$$(1-t^2)y^{″}+(\beta -\alpha -(\alpha +\beta +2)t)y^{\prime }+\nu (\nu +\alpha +\beta +1)y=0.$$

Jacobi polynomials can alternatively be characterized by the following recursive relationship:

$$P_{\nu }^{(\alpha ,\beta )}\left(t\right)=(A_{\nu -1}z-B_{\nu -1})P_{\nu -1}^{(\alpha ,\beta )}\left(t\right)-C_{\nu -1}{P}_{\nu -2}^{(\alpha ,\beta )}\left(t\right),\nu ⩾2,$$

(3)

where

$A_{\nu }={\displaystyle \frac{\left(2\nu +\alpha +\beta +1\right)\left(2\nu +\alpha +\beta +2\right)}{2(\nu +1)\left(\nu +\alpha +\beta +1\right)}}$, $B_{\nu }={\displaystyle \frac{\left(2\nu +\alpha +\beta +1\right)\left({\beta }^2-{\alpha }^2\right)}{2(\nu +1)\left(\nu +\alpha +\beta +1\right)(2\nu +\alpha +\beta )}}$, and $C_{\nu }={\displaystyle \frac{\left(2\nu +\alpha +\beta +2\right)\left(\nu +\alpha \right)\left(\nu +\beta \right)}{(\nu +1)\left(\nu +\alpha +\beta +1\right)(2\nu +\alpha +\beta )}}$, with the initial values

$$P_0^{(\alpha ,\beta )}\left(t\right)=1,P_1^{(\alpha ,\beta )}\left(t\right)=\left(\alpha +1\right)+{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)$$

(4)

and

$$\begin{array}{c}\hfill P_2^{(\alpha ,\beta )}\left(t\right)={\displaystyle \frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}}+{\displaystyle \frac{1}{2}}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)+{\displaystyle \frac{1}{8}}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2.\end{array}$$

To begin, we present specific cases of the polynomials $P_{\nu }^{(\alpha ,\beta )}$. When $\alpha =\beta =0$, the polynomials reduce to the Legendre polynomials. Setting $\alpha =\beta =-0.5$ results in the Chebyshev polynomials of the first kind, while $\alpha =\beta =0.5$ yields the Chebyshev polynomials of the second kind. Moreover, when $\alpha =\beta$, the polynomials simplify to Gegenbauer polynomials, with $\alpha$ replaced by $\left(\alpha -0.5\right)$.

Let $\mathcal{U}$ be the family of analytic and univalent functions L in the open unit disk $\mathbb{D}$ of the form

$$L\left(z\right)=z+c_2{z}^2+c_3{z}^3+\cdots ,$$

(5)

Differential subordination, introduced by Miller and Mocanu [8], is a key framework in geometric function theory. Their pioneering work laid the groundwork for subsequent studies and applications of differential subordination of analytic functions. Furthermore, their book [9] provides a comprehensive overview of advancements and references in the field up to its publication.

For analytic functions L and V, L subordination to V (denoted by $L\prec V$ ) for all $z\in \mathbb{D}$, if there exists a function $\varpi$ via $\varpi \left(0\right)=0$ and $\left|\varpi \right(z\left)\right|<1$, such that

$$L\left(z\right)=V\left(\varpi \right(z\left)\right).$$

In addition [10], if V is univalent in $\mathbb{D}$, then

$$L\left(z\right)\prec V\left(z\right)\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}L\left(0\right)=V\left(0\right)$$

and

$$L\left(\mathbb{D}\right)\subset V\left(\mathbb{D}\right).$$

Every function $L\in \mathcal{U}$ has an inverse $L^{-1}$ defined by (see [11])

$$L^{-1}\left(L\left(z\right)\right)=z(z\in \mathbb{D})$$

and

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

where

$$V\left(\varpi \right):=L^{-1}\left(\varpi \right)=\varpi -c_2{\varpi }^2+(2c_2^2-c_3){\varpi }^3-(c_4+5c_2^3-5c_3{c}_2){\varpi }^4+\cdots .$$

(6)

A function $L\in \mathcal{U}$ is said to be bi-univalent in $\mathbb{D}$ (the family of bi-univalent functions in $\mathbb{D}$ denoted by $\Sigma )$ if both $L\left(z\right)$ and $L^{-1}\left(z\right)$ are univalent in $\mathbb{D}$ (see [12,13,14,15,16]).

The study of bi-univalent functions is crucial in geometric function theory, particularly in exploring functions that are univalent in both a domain and its inverse. Their connection with orthogonal polynomials, like Jacob polynomials, aids in analyzing coefficient bounds and structural properties. Applications span diverse fields, including low-light imaging for enhanced contrast, image edge detection for precision, and stealth combat aircraft for radar signature optimization (see [17,18,19]).

The error function plays a significant role in various scientific fields, including probability, statistics, partial differential equations, and other engineering applications. Consequently, it has garnered substantial attention in mathematics. Numerous studies have explored inequalities and related properties of the error function; for instance, see [20,21,22,23]. Additionally, the error function and its approximations are widely used in predicting events with very high or very low probabilities.

The error function, denoted by the symbol $erf$, is defined by [24]

$$erf\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }}}\underset{0}{\stackrel{z}{\int }}{e}^{-t^2}dt={\displaystyle \frac{2}{\sqrt{\pi }}}\sum _{\nu =0}^{\infty }{\displaystyle \frac{{(-1)}^{\nu }{z}^{2\nu +1}}{(2\nu +1)\nu !}},z\in \mathbb{C}.$$

(7)

The error function $erf$ over real numbers and in the complex plane is represented graphically in Figure 1 and Figure 2, respectively.

As illustrated above, the series is derived by expanding the integrand $e^{-t^2}$ into its Maclaurin series and integrating term by term. Moreover, the Maclaurin series of the imaginary error function $erfi$ is very similar, as explained by (see [26,27])

$$erfi\left(z\right)={\displaystyle \frac{2}{\sqrt{\pi }}}\sum _{\nu =0}^{\infty }{\displaystyle \frac{z^{2\nu +1}}{(2\nu +1)\nu !}},z\in \mathbb{C}.$$

(8)

The imaginary error function $erfi\left(z\right)$ in the complex plane is represented graphically in Figure 3.

Using (7), Ramachandran et al. [28] investigated the normalized analytic error function regarding the form

$$\mathrm{Erf}\left(z\right)={\displaystyle \frac{\sqrt{\pi z}}{2}}erf\left(\sqrt{z}\right)=z+\sum _{\nu =2}^{\infty }{\displaystyle \frac{{(-1)}^{\nu -1}{z}^{\nu }}{(2\nu -1)(\nu -1)!}}.$$

(9)

The convolution product “∗” is an operation that combines two power series [29], where the coefficients of the resulting series are the products of the coefficients from the original series. Formally, the product of two series $f\left(z\right)={\displaystyle \sum _{\nu =2}^{\infty }}{a}_{\nu }{z}^{\nu }$ and $g\left(z\right)={\displaystyle \sum _{\nu =2}^{\infty }}{b}_{\nu }{z}^{\nu }$ results in a new series $h\left(z\right)={\displaystyle \sum _{\nu =2}^{\infty }}\left(a_{\nu }{b}_{\nu }\right)z^{\nu }$, utilizing the convolution product defined the following family:

$$\mathrm{Erf}\ast \mathcal{U}=\left\{G:G\left(z\right)=(\mathrm{Erf}\ast L)\left(z\right)=z+\sum _{\nu =2}^{\infty }{\displaystyle \frac{{(-1)}^{\nu -1}{c}_{\nu }}{(2\nu -1)(\nu -1)!}}{z}^{\nu },L\in \mathcal{U}\right\}.$$

(10)

From (8), the normalized analytic imaginary error function Erfi is defined by

$$\mathrm{Erfi}\left(z\right)={\displaystyle \frac{\sqrt{\pi z}}{2}}erfi\left(\sqrt{z}\right)=z+\sum _{\nu =2}^{\infty }{\displaystyle \frac{z^{\nu }}{(2\nu -1)(\nu -1)!}},$$

and, by the convolution product, we define

$$EL\left(z\right)=(\mathrm{Erfi}\ast L)\left(z\right)=z+\sum _{\nu =2}^{\infty }{\displaystyle \frac{c_{\nu }}{(2\nu -1)(\nu -1)!}}{z}^{\nu },L\in \mathcal{U}.$$

2. The Family ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$

After we defined the Jacobi polynomials and the normalized analytic imaginary error function, we will introduce the following subfamily of bi-univalent functions.

Definition 1.

A function $L\in \Sigma$ given by (5) is said to be in the family ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$ if it satisfies the two conditions below:

$$(1-\tau ){\displaystyle \frac{EL\left(z\right)}{z}}+\tau {\left(EL\left(z\right)\right)}^{\prime }+\lambda z{\left(EL\left(z\right)\right)}^{″}\prec J_{\nu }(t,z)$$

(11)

and

$$(1-\tau ){\displaystyle \frac{EV\left(\varpi \right)}{\varpi }}+\tau {\left(EV\left(\varpi \right)\right)}^{\prime }+\lambda \varpi {\left(EV\left(\varpi \right)\right)}^{″}\prec J_{\nu }(t,\varpi ),$$

(12)

where $z,\varpi \in \mathbb{D}$, $\tau ,\lambda \ge 0$, $\alpha ,\beta >-1$, $t\in ({\displaystyle \frac{1}{2}},1]$, and the function $V=L^{-1}$ is given by (6).

To establish that the family ${\mathfrak{J}}_{\Sigma }$ contains non-trivial functions, we refer the reader to [30], where a rigorous proof is provided. In particular, ref. [30] demonstrates the construction of an explicit example of a function that belongs to ${\mathfrak{J}}_{\Sigma }$, thereby ensuring that the family is not empty.

Subfamily 1.

For $\lambda =0$, we have that ${\mathfrak{J}}_{\Sigma }(t,0,\tau ):={\mathfrak{J}}_{\Sigma }(t,\tau )$ is the subfamily of functions $L\in \Sigma$ that satisfies the below conditions:

$$(1-\tau ){\displaystyle \frac{EL\left(z\right)}{z}}+\tau {\left(EL\left(z\right)\right)}^{\prime }\prec J_{\nu }(t,z)$$

and

$$(1-\tau ){\displaystyle \frac{EV\left(\varpi \right)}{\varpi }}+\tau {\left(EV\left(\varpi \right)\right)}^{\prime }\prec J_{\nu }(t,\varpi ),$$

where $z,\varpi \in \mathbb{D}$, $\tau \ge 0$, $\alpha ,\beta >-1$, and $t\in ({\displaystyle \frac{1}{2}},1]$.

Subfamily 2.

For $\lambda =0$ and $\tau =1$, we have that ${\mathfrak{J}}_{\Sigma }(t,0,1):={\mathfrak{J}}_{\Sigma }(t,1)$ is the subfamily of functions $L\in \Sigma$, satisfying the below conditions:

$${\left(EL\left(z\right)\right)}^{\prime }\prec J_{\nu }(t,z)$$

and

$${\left(EV\left(\varpi \right)\right)}^{\prime }\prec J_{\nu }(t,\varpi ),$$

where $z,\varpi \in \mathbb{D}$, $\alpha ,\beta >-1$, and $t\in ({\displaystyle \frac{1}{2}},1]$.

Subfamily 3.

For $\lambda =0$ and $\tau =0$, we have that ${\mathfrak{J}}_{\Sigma }(t,0,0):={\mathfrak{J}}_{\Sigma }\left(t\right)$ is subfamily of functions $L\in \Sigma$ that satisfies the below conditions:

$${\displaystyle \frac{EL\left(z\right)}{z}}\prec J_{\nu }(t,z)$$

and

$${\displaystyle \frac{EV\left(\varpi \right)}{\varpi }}\prec J_{\nu }(t,\varpi ),$$

where $z,\varpi \in \mathbb{D}$, $\alpha ,\beta >-1$, and $t\in ({\displaystyle \frac{1}{2}},1]$.

Recently, many researchers have examined bi-univalent functions associated with orthogonal polynomials, obtaining non-sharp estimates for the Maclaurin coefficients $|c_2|$ and $|c_3|$ (see [31,32,33,34,35,36,37]). In addition, numerous studies in recent years have employed various special functions, such as Borel, Poisson, Rabotnov, Pascal, Wright, and Bessel, to investigate key aspects of geometric function theory, including coefficient estimates, inclusion relations, and criteria for membership in specific families (see [38,39,40,41,42,43,44]).

The content of this paper is organized as follows. In Section 3, we provide bounds for the coefficients $|c_2|$ and $|c_3|$ in the Maclaurin expansions, along with an estimation of the Fekete–Szegő inequality for functions in the family ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$. Section 4 highlights pertinent links between certain particular cases of the main results. Finally, Section 5 concludes the study with a few observations.

3. Bounds of the Family of Bi-Univalent Functions ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$

Section 3 begins by providing bounds for the coefficients $|c_2|$ and $|c_3|$ in the Maclaurin expansions for functions in the family ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$.

Theorem 1.

Let $L\in \Sigma$, as given by (5), belong to the family ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$. Then,

$$\left|c_2\right|\le {\displaystyle \frac{\left(\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right)\sqrt{2\left(\alpha +1\right)+(\alpha +\beta +2)(t-1)}}{\sqrt{\left|{\rm Y}(t,\alpha ,\lambda ,\beta )\right|}}}$$

and

$$\left|c_3\right|\le {\displaystyle \frac{9{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}{{\left(2\lambda +\tau +1\right)}^2}}+{\displaystyle \frac{10\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}{6\lambda +2\tau +1}},$$

where

$$\begin{array}{ccc}\hfill {\rm Y}(t,\alpha ,\lambda ,\beta )& =& {\displaystyle \frac{1}{5}}\left(6\lambda +2\tau +1\right){\left[\begin{array}{c}\left(\alpha +1\right)\\ +{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\end{array}\right]}^2\hfill \\ & & -{\displaystyle \frac{2}{9}}{\left(2\lambda +\tau +1\right)}^2\left[\begin{array}{c}{\displaystyle \frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}}+{\displaystyle \frac{1}{2}}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +{\displaystyle \frac{1}{8}}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right].\hfill \end{array}$$

Proof. 

Let $L\in {\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$. From Definition 1, we can write

$$(1-\tau ){\displaystyle \frac{EL\left(z\right)}{z}}+\tau {\left(EL\left(z\right)\right)}^{\prime }+\lambda z{\left(EL\left(z\right)\right)}^{″}=J_{\nu }(t,p\left(z\right))$$

(13)

and

$$(1-\tau ){\displaystyle \frac{EV\left(\varpi \right)}{\varpi }}+\tau {\left(EV\left(\varpi \right)\right)}^{\prime }+\lambda \varpi {\left(EV\left(\varpi \right)\right)}^{″}=J_{\nu }(t,q\left(\varpi \right)),$$

(14)

where p and q are analytic and have the form

$$p\left(z\right)=j_{1}z+j_2{z}^2+j_3{z}^3+\cdots ,(z\in \mathbb{D})$$

and

$$q\left(\varpi \right)=d_1\varpi +d_2{\varpi }^2+d_3{\varpi }^3+\cdots ,(\varpi \in \mathbb{D}),$$

such that $p\left(0\right)=q\left(0\right)=0$ and $\left|p\right(z\left)\right|<1$, $\left|q\right(\varpi \left)\right|<1$ for all $z,\varpi \in \mathbb{D}$.

From Equalities (13) and (14), we obtain

$$(1-\tau ){\displaystyle \frac{EL\left(z\right)}{z}}+\tau E{L}^{\prime }\left(z\right)+\lambda zE{L}^{″}\left(z\right)=1+P_1^{(\alpha ,\beta )}\left(t\right)j_{1}z+\left[\begin{array}{c}{P}_1^{(\alpha ,\beta )}\left(t\right)j_2\\ +P_2^{(\alpha ,\beta )}\left(t\right)j_1^2\end{array}\right]z^2+\cdots$$

(15)

and

$$(1-\tau ){\displaystyle \frac{EV\left(\varpi \right)}{\varpi }}+\tau E{V}^{\prime }\left(\varpi \right)+\lambda \varpi E{V}^{″}\left(\varpi \right)=1+P_1^{(\alpha ,\beta )}\left(t\right)d_1\varpi +\left[\begin{array}{c}{P}_1^{(\alpha ,\beta )}\left(t\right)d_2\\ +P_2^{(\alpha ,\beta )}\left(t\right)d_1^2\end{array}\right]{\varpi }^2+\cdots .$$

(16)

As is widely known, if

$$\left|p\left(z\right)\right|=\left|j_{1}z+j_2{z}^2+j_3{z}^3+\cdots \right|<1,(z\in \mathbb{D})$$

and

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

then

$$|j_i|\le 1\mathrm{and}|d_i|\le 1\mathrm{for}\mathrm{all}i\in \mathbb{N}.$$

(17)

Equating the coefficients of both sides in (15) and (16), we obtain

$${\displaystyle \frac{1}{3}}\left(2\lambda +\tau +1\right)c_2=P_1^{(\alpha ,\beta )}\left(t\right)j_1,$$

(18)

$${\displaystyle \frac{1}{10}}\left(6\lambda +2\tau +1\right)c_3=P_1^{(\alpha ,\beta )}\left(t\right)j_2+P_2^{(\alpha ,\beta )}\left(t\right)j_1^2,$$

(19)

$$-{\displaystyle \frac{1}{3}}\left(2\lambda +\tau +1\right)c_2=P_1^{(\alpha ,\beta )}\left(t\right)d_1,$$

(20)

and

$${\displaystyle \frac{1}{10}}\left(6\lambda +2\tau +1\right)\left[2c_2^2-c_3\right]=P_1^{(\alpha ,\beta )}\left(t\right)d_2+P_2^{(\alpha ,\beta )}\left(t\right)d_1^2.$$

(21)

It follows from (18) and (20) that

$$j_1=-d_1$$

(22)

and

$${\displaystyle \frac{2}{9}}{\left(2\lambda +\tau +1\right)}^2{c}_2^2={\left[P_1^{(\alpha ,\beta )}\left(t\right)\right]}^2\left(j_1^2+d_1^2\right).$$

(23)

If we add (19) and (21), we obtain

$${\displaystyle \frac{1}{5}}\left(6\lambda +2\tau +1\right)c_2^2=P_1^{(\alpha ,\beta )}\left(t\right)\left(j_2+d_2\right)+P_2^{(\alpha ,\beta )}\left(t\right)\left(j_1^2+d_1^2\right).$$

(24)

Replacing the value of $\left(c_1^2+d_1^2\right)$ from (23) in the right hand side of (24), we have

$$\left[{\displaystyle \frac{1}{5}}\left(6\lambda +2\tau +1\right)-{\displaystyle \frac{2}{9}}{\left(2\lambda +\tau +1\right)}^2{\displaystyle \frac{P_2^{(\alpha ,\beta )}\left(t\right)}{{\left[P_1^{(\alpha ,\beta )}\left(t\right)\right]}^2}}\right]c_2^2=P_1^{(\alpha ,\beta )}\left(t\right)\left(j_2+d_2\right).$$

(25)

Using (4) and (17) in (25), we find that

$$\left|c_2\right|\le {\displaystyle \frac{\left(\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right)\sqrt{2\left(\alpha +1\right)+(\alpha +\beta +2)(t-1)}}{\sqrt{\left|{\rm Y}(t,\alpha ,\lambda ,\beta )\right|}}},$$

where

$$\begin{array}{ccc}\hfill {\rm Y}(t,\alpha ,\lambda ,\beta )& =& {\displaystyle \frac{1}{5}}\left(6\lambda +2\tau +1\right){\left[\begin{array}{c}\left(\alpha +1\right)\\ +{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\end{array}\right]}^2\hfill \\ & & -{\displaystyle \frac{2}{9}}{\left(2\lambda +\tau +1\right)}^2\left[\begin{array}{c}{\displaystyle \frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}}+{\displaystyle \frac{1}{2}}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +{\displaystyle \frac{1}{8}}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right].\hfill \end{array}$$

Also, if we subtract (21) from (19), we obtain

$${\displaystyle \frac{1}{5}}\left(6\lambda +2\tau +1\right)\left(c_3-c_2^2\right)=P_1^{(\alpha ,\beta )}\left(t\right)\left(j_2-d_2\right)+P_2^{(\alpha ,\beta )}\left(t\right)\left(j_1^2-d_1^2\right).$$

(26)

Then, from (22) and (23), Equation (26) becomes

$$c_3={\displaystyle \frac{9{\left[P_1^{(\alpha ,\beta )}\left(t\right)\right]}^2}{2{\left(2\lambda +\tau +1\right)}^2}}\left(j_1^2+d_1^2\right)+{\displaystyle \frac{5P_1^{(\alpha ,\beta )}\left(t\right)}{6\lambda +2\tau +1}}\left(j_2-d_2\right).$$

By applying (4), we conclude that

$$\left|c_3\right|\le {\displaystyle \frac{9{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}{{\left(2\lambda +\tau +1\right)}^2}}+{\displaystyle \frac{10\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}{6\lambda +2\tau +1}}.$$

□

Using the values of $c_2$ and $c_3$, we estimate the functional $\left|c_3-\phi c_2^2\right|$ for functions in the family of bi-univalent functions ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$.

Theorem 2.

Let $L\in \Sigma$, as given by (5), belong to the family ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$. Then,

$$\left|c_3-\phi c_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{10\left|\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right|}{6\lambda +2\tau +1}}\\ \\ {\displaystyle \frac{2{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^3\left|1-\phi \right|}{\left|{\rm Y}(t,\alpha ,\lambda ,\beta )\right|}}\end{array}\right.\begin{array}{c}\left|1-\phi \right|\le {\Pi }_1,\\ \\ \left|1-\phi \right|\ge {\Pi }_1,\end{array}$$

where

$${\Pi }_1=1-{\displaystyle \frac{\frac{10}{9}{\left(2\lambda +\tau +1\right)}^2\left(\begin{array}{c}\frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}+\frac{1}{2}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +\frac{1}{8}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right)}{\left(6\lambda +2\tau +1\right){\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}}.$$

Proof. 

From (25) and (26), we have

$$\begin{array}{cc}\hfill c_3-\phi c_2^2& ={\displaystyle \frac{5P_1^{(\alpha ,\beta )}\left(t\right)}{6\lambda +2\tau +1}}\left(j_2-d_2\right)\hfill \\ \hfill & +\left(1-\phi \right){\displaystyle \frac{{\left[P_1^{(\alpha ,\beta )}\left(t\right)\right]}^3\left(j_2+d_2\right)}{\frac{1}{5}\left(6\lambda +2\tau +1\right){\left[P_1^{(\alpha ,\beta )}\left(t\right)\right]}^2-\frac{2}{9}{\left(2\lambda +\tau +1\right)}^2{P}_2^{(\alpha ,\beta )}\left(t\right)}}\hfill \\ \hfill & =P_1^{(\alpha ,\beta )}\left(t\right)\left[ϝ\left(\phi \right)+{\displaystyle \frac{5}{6\lambda +2\tau +1}}\right]j_2+P_1^{(\alpha ,\beta )}\left(t\right)\left[ϝ\left(\phi \right)-{\displaystyle \frac{5}{6\lambda +2\tau +1}}\right]d_2,\hfill \end{array}$$

where

$$ϝ\left(\phi \right)={\displaystyle \frac{{\left[P_1^{(\alpha ,\beta )}\left(t\right)\right]}^2\left(1-\phi \right)}{\frac{1}{5}\left(6\lambda +2\tau +1\right){\left[P_1^{(\alpha ,\beta )}\left(t\right)\right]}^2-\frac{2}{9}{\left(2\lambda +\tau +1\right)}^2{P}_2^{(\alpha ,\beta )}\left(t\right)}}.$$

Then, from (4), we deduce that

$$\left|c_3-\phi c_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{10\left|h_2\left(y\right)\right|}{6\lambda +2\tau +1}}\\ \\ 2\left|P_1^{(\alpha ,\beta )}\left(t\right)\right|\left|ϝ\left(\phi \right)\right|\end{array}\right.\begin{array}{c}\left|ϝ\left(\phi \right)\right|\le {\displaystyle \frac{5}{6\lambda +2\tau +1}},\\ \\ \left|ϝ\left(\phi \right)\right|\ge {\displaystyle \frac{5}{6\lambda +2\tau +1}}.\end{array}$$

$$\equiv \left\{\begin{array}{c}{\displaystyle \frac{10\left|\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right|}{6\lambda +2\tau +1}}\\ \\ {\displaystyle \frac{2{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^3\left|1-\phi \right|}{\left|{\rm Y}(t,\alpha ,\lambda ,\beta )\right|}}\end{array}\right.\begin{array}{c}\left|1-\phi \right|\le {\Pi }_1,\\ \\ \left|1-\phi \right|\ge {\Pi }_1,\end{array}$$

where

$${\Pi }_1=1-{\displaystyle \frac{\frac{10}{9}{\left(2\lambda +\tau +1\right)}^2\left(\begin{array}{c}\frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}+\frac{1}{2}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +\frac{1}{8}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right)}{\left(6\lambda +2\tau +1\right){\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}}.$$

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4. Particular Cases

The following corollaries are obtained by specializing parameters $\lambda$ and $\tau$ in the aforementioned theorems in Section 3.

Corollary 1.

Let $L\in \Sigma$, as given by (5), belong to the family${\mathfrak{J}}_{\Sigma }(t,\tau )$. Then,

$$\left|c_2\right|\le {\displaystyle \frac{\left(\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right)\sqrt{2\left(\alpha +1\right)+(\alpha +\beta +2)(t-1)}}{\sqrt{\left|{\rm Y}(t,\alpha ,0,\beta )\right|}}},$$

$$\left|c_3\right|\le {\displaystyle \frac{9{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}{{\left(\tau +1\right)}^2}}+{\displaystyle \frac{10\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}{2\tau +1}},$$

and

$$\left|c_3-\phi c_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{10\left|\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right|}{2\tau +1}}\\ \\ {\displaystyle \frac{2{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^3\left|1-\phi \right|}{\left|{\rm Y}(t,\alpha ,\beta )\right|}}\end{array}\right.\begin{array}{c}\left|1-\phi \right|\le {\Pi }_2,\\ \\ \left|1-\phi \right|\ge {\Pi }_2,\end{array}$$

where

$$\begin{array}{ccc}\hfill {\rm Y}(t,\alpha ,\beta )& =& {\displaystyle \frac{1}{5}}\left(2\tau +1\right){\left[\begin{array}{c}\left(\alpha +1\right)\\ +{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\end{array}\right]}^2\hfill \\ & & -{\displaystyle \frac{2}{9}}{\left(\tau +1\right)}^2\left[\begin{array}{c}{\displaystyle \frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}}+{\displaystyle \frac{1}{2}}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +{\displaystyle \frac{1}{8}}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right]\hfill \end{array}$$

and

$${\Pi }_2=1-{\displaystyle \frac{\frac{10}{9}{\left(\tau +1\right)}^2\left(\begin{array}{c}\frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}+\frac{1}{2}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +\frac{1}{8}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right)}{\left(2\tau +1\right){\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}}.$$

Corollary 2.

Let $L\in \Sigma$, as given by (5), belong to the family${\mathfrak{J}}_{\Sigma }(t,1)$. Then,

$$\left|c_2\right|\le {\displaystyle \frac{\left(\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right)\sqrt{2\left(\alpha +1\right)+(\alpha +\beta +2)(t-1)}}{\sqrt{\left|{\rm Y}(t,\alpha ,\beta )\right|}}},$$

$$\left|c_3\right|\le {\displaystyle \frac{9{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}{4}}+{\displaystyle \frac{10\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}{3}},$$

and

$$\left|c_3-\phi c_2^2\right|\le \left\{\begin{array}{c}{\displaystyle \frac{10\left|\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right|}{3}}\\ \\ {\displaystyle \frac{2{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^3\left|1-\phi \right|}{\left|{\rm Y}(t,\alpha ,\beta )\right|}}\end{array}\right.\begin{array}{c}\left|1-\phi \right|\le {\Pi }_3,\\ \\ \left|1-\phi \right|\ge {\Pi }_3,\end{array}$$

where

$$\begin{array}{ccc}\hfill {\rm Y}(t,\alpha ,\lambda ,\beta )& =& {\displaystyle \frac{3}{5}}{\left[\begin{array}{c}\left(\alpha +1\right)\\ +{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\end{array}\right]}^2\hfill \\ & & -{\displaystyle \frac{8}{9}}\left[\begin{array}{c}{\displaystyle \frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}}+{\displaystyle \frac{1}{2}}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +{\displaystyle \frac{1}{8}}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right]\hfill \end{array}$$

and

$${\Pi }_3=1-{\displaystyle \frac{\frac{40}{9}\left(\begin{array}{c}\frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}+\frac{1}{2}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +\frac{1}{8}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right)}{3{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}}.$$

Corollary 3.

Let $L\in \Sigma$, as given by (5), belong to the family${\mathfrak{J}}_{\Sigma }\left(t\right)$. Then,

$$\left|c_2\right|\le {\displaystyle \frac{\left(\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right)\sqrt{2\left(\alpha +1\right)+(\alpha +\beta +2)(t-1)}}{\sqrt{\left|{\rm Y}(t,\alpha ,\beta )\right|}}},$$

$$\left|c_3\right|\le 9{\left[\left(\alpha +1\right)+{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\right]}^2+10\left[\left(\alpha +1\right)+{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\right],$$

and

$$\left|c_3-\phi c_2^2\right|\le \left\{\begin{array}{c}10\left|\left(\alpha +1\right)+{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\right|\\ \\ {\displaystyle \frac{2{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^3\left|1-\phi \right|}{\left|{\rm Y}(t,\alpha ,\beta )\right|}}\end{array}\right.\begin{array}{c}\left|1-\phi \right|\le {\Pi }_4,\\ \\ \left|1-\phi \right|\ge {\Pi }_4,\end{array}$$

where

$$\begin{array}{ccc}\hfill {\rm Y}(t,\alpha ,\beta )& =& {\displaystyle \frac{1}{5}}{\left[\begin{array}{c}\left(\alpha +1\right)\\ +{\displaystyle \frac{1}{2}}(\alpha +\beta +2)(t-1)\end{array}\right]}^2\hfill \\ & & -{\displaystyle \frac{2}{9}}\left[\begin{array}{c}{\displaystyle \frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}}+{\displaystyle \frac{1}{2}}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +{\displaystyle \frac{1}{8}}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right]\hfill \end{array}$$

and

$${\Pi }_4=1-{\displaystyle \frac{\frac{10}{9}\left(\begin{array}{c}\frac{\left(\alpha +1\right)\left(\alpha +2\right)}{2}+\frac{1}{2}\left(\alpha +2\right)(\alpha +\beta +3)(t-1)\\ +\frac{1}{8}(\alpha +\beta +3)(\alpha +\beta +4){(t-1)}^2\end{array}\right)}{{\left[\left(\alpha +1\right)+\frac{1}{2}(\alpha +\beta +2)(t-1)\right]}^2}}.$$

5. Conclusions

In this paper, we defined a comprehensive family of analytic and bi-univalent functions associated with the imaginary error function and subordinate to Jacobi polynomials, denoted by ${\mathfrak{J}}_{\Sigma }(t,\lambda ,\tau )$. We provided estimates for the Maclaurin coefficients $\left|c_2\right|$, $\left|c_3\right|$, and addressed the Fekete–Szegő problems. Furthermore, by specializing the parameters $t,\lambda$, and $\tau$, the results for the subfamilies ${\mathfrak{J}}_{\Sigma }(t,\tau )$, ${\mathfrak{J}}_{\Sigma }(t,1)$, and ${\mathfrak{J}}_{\Sigma }\left(t\right)$ can be derived.

The use of the normalized error function may inspire researchers to determine coefficient estimates, such as $\left|c_2\right|$, $\left|c_3\right|$, and the Fekete–Szegő problems for functions, belonging to new subfamily of bi-univalent functions. Additionally, we anticipate that this study will encourage other researchers to extend this family to harmonic functions and symmetric q-calculus. The approach can also be adapted to utilize the symmetric q-sine and q-cosine domains as alternatives to the current domain.