Coefficients and Fekete–Szegö Functional Estimations of Bi-Univalent Subclasses Based on Gegenbauer Polynomials

Abstract

Subclasses of analytic and bi-univalent functions have been extensively improved and utilized for estimating the Taylor–Maclaurin coefficients and the Fekete–Szegö functional. In this paper, we consider a certain subclass of normalized analytic and bi-univalent functions. These functions have inverses that possess a bi-univalent analytic continuation to an open unit disk and are associated with orthogonal polynomials; namely, Gegenbauer polynomials that satisfy subordination conditions on the open unit disk. We use this subclass to derive new approximations for the second and third Taylor–Maclaurin coefficients and the Fekete–Szegö functional. Furthermore, we discuss several new results that arise when we specialize the parameters used in our fundamental findings.

1. Introduction

Gegenbauer polynomials (GPs), denoted by $C_n\left(x\right)$, are orthogonal polynomials on the interval $I=[-1,1]$ with respect to the weight function ${(1-x^2)}^{\alpha -\frac{1}{2}}$, where $\alpha >-\frac{1}{2}$. These polynomials can be recursively defined as follows:

$$\begin{array}{cc}\hfill & C_0^{\left(\alpha \right)}\left(x\right)=1,\hfill \\ \hfill & C_1^{\left(\alpha \right)}\left(x\right)=2\alpha x,\hfill \\ \hfill & C_2^{\left(\alpha \right)}\left(x\right)=2\alpha (1+\alpha )x^2-\alpha ,\hfill \\ \hfill & C_n^{\left(\alpha \right)}\left(x\right)=\frac{1}{n}\left[2x\left(n+\alpha -1\right)C_{n-1}^{\left(\alpha \right)}\left(x\right)-\left(n+2\alpha -2\right)C_{n-2}^{\left(\alpha \right)}\left(x\right)\right],n\ge 2.\hfill \end{array}$$

(1)

That is, for any two GPs, $C_n^{\left(\alpha \right)}\left(x\right)$ and $C_m^{\left(\alpha \right)}\left(x\right)$, with $n\ne m$, we have

$${\int }_{-1}^1{C}_n^{\left(\alpha \right)}\left(x\right)C_m^{\left(\alpha \right)}\left(x\right){\left(1-x^2\right)}^{\alpha -\frac{1}{2}}dx=0,$$

and with the case that $n=m$, we have

$${\int }_{-1}^1{\left(C_n^{\left(\alpha \right)}\left(x\right)\right)}^2{\left(1-x^2\right)}^{\alpha -\frac{1}{2}}dx=\frac{\sqrt{\pi }\Gamma (n+2\alpha -1)}{2^{1-2\alpha }n!\Gamma \left(\alpha \right)\Gamma (n+\alpha -\frac{1}{2})}.$$

The GPs have a generating function, $H_{\alpha }(x,z)$, that is given by the following decomposition

$$H_{\alpha }(x,z)=\frac{1}{{\left(1-2xz+z^2\right)}^{\alpha }}=\sum _{n=0}^{\infty }{C}_n^{\left(\alpha \right)}\left(x\right)z^n\hspace{1em}\alpha >0,$$

(2)

where $x\in I$ and z are in the open unit disk $\mathbb{D}=\{z:z\in \mathbb{C}\mathrm{and}\left|z\right|<1\}$, and $\mathbb{C}$ is, as usual, the set of complex numbers. For a fixed $x\in I$, $H_{\alpha }(x,z)$ is analytic in $\mathbb{D}$ that can be expanded in a Taylor series as defined in $\left(2\right)$. Note that when $\alpha =0$, $H_{\alpha }(x,z)$ produces no values, and therefore the generating function of the GP is defined by

$$H_{{}_0}(x,z)=1-log\left(1-2xz+z^2\right)=\sum _{n=0}^{\infty }{C}_n^{\left(0\right)}\left(x\right)z^n.$$

(3)

The GP of degree n is a particular solution to the Gegenbauer differential equation given by

$$(1-x^2)\frac{d^{2}y}{d{x}^2}-(2\alpha +1)x\frac{dy}{dx}+n(n+2\alpha )y=0.$$

(4)

Note that by setting $\alpha =\frac{1}{2}$ and $\alpha =1$ in the above equation, it reduces to the Legendre and Chebyshv differential equations and the GPs reduce to Legendre polynomials (LP’s) and Chebyshv polynomials (CPs) of the second type, respectively.

Let $\mathcal{A}$ denote the class of all analytic functions f defined in $\mathbb{D}$ and normalized by the conditions $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1$. Thus, each $f\in \mathcal{A}$ has a Taylor–Maclaurin series expansion of the form

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

(5)

Moreover, let $\mathcal{S}$ denote the class of all functions $f\in \mathcal{A}$ that are univalent in $\mathbb{D}$. Two functions, f and g, are said to be subordinate, written as $f\prec g$, if an analytic function $h\left(z\right)$ (called a Schwarz function) is found in $\mathbb{D}$, such that $f\left(z\right)=g\left(h\right(z\left)\right)$ with $h\left(0\right)=0$ and $\left|h\right(z\left)\right|\le 1$. In particular, if the function g is univalent in $\mathbb{D}$, then the following equivalence is valid [1]

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

and

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

By the Koebe one-quarter theorem [2], the image of $\mathbb{D}$ under every function $f\in \mathcal{S}$ will contain the disk of radius $\frac{1}{4}$ and centre at the origin, i.e., ${\mathbb{D}}_{\frac{1}{4}}\left(0\right)\in f\left(\mathbb{D}\right)$. According to this, one can see that every function $f\in \mathcal{S}$ has an inverse $f^{-1}:f\left(\mathbb{D}\right)\to \mathbb{D}$ satisfying the following conditions

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

and

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

where in fact $f^{-1}$ has the series expansion of the form

$$f^{-1}\left(w\right)=w-a_{{}_2}{w}^2+(2a_{{}_2}^2-a_{{}_3})w^3-(5a_{{}_2}^3-5a_{{}_2}{a}_{{}_3}+a_{{}_4})w^4+\cdots .$$

(6)

Note that a function $f\in \mathcal{A}$ is said to be bi-univalent in $\mathbb{D}$ if both the function, f, and its inverse, $f^{-1}$, are univalent in $\mathbb{D}$. Let $\mathcal{F}$ denotes the class of bi-univalent functions in $\mathbb{D}$ given by $\left(5\right)$; for example, the following functions

$$f_{{}_1}\left(z\right)=\frac{z}{1-z}\hspace{1em}{f}_{{}_2}\left(z\right)=-\log (1-z)\hspace{1em}\mathrm{and}\hspace{1em}{f}_{{}_3}\left(z\right)=\frac{1}{2}\log \left(\frac{1+z}{1-z}\right),$$

with their respective inverses

$$f_{{}_1}^{-1}\left(w\right)=\frac{w}{1+w}\hspace{1em}{f}_{{}_2}^{-1}\left(w\right)=\frac{e^w-1}{e^w}\hspace{1em}\mathrm{and}\hspace{1em}{f}_{{}_3}^{-1}\left(w\right)=\frac{e^{2w}-1}{e^{2w}+1},$$

are bi-univalent. However, the Koebe function, $K\left(z\right)=\frac{z}{{(1-z)}^2}$, is not a member of the class $\mathcal{F}$ since it maps the open unit disk $\mathbb{D}\subset \mathbb{C}$ onto $\mathrm{K}\left(\mathbb{D}\right)=\mathbb{C}\setminus (-\infty ,-\frac{1}{4}]$, which does not contain $\mathbb{D}$ (i.e., $\{w:w\in \mathbb{C}\mathrm{and}|w|\le \frac{1}{4}\}\subseteq \mathrm{K}\left(\mathbb{D}\right)$) (see [3,4,5,6,7,8,9,10]). Other common univalent functions that are not members of $\mathcal{F}$ are

$$g\left(z\right)=\frac{z}{1-z^2},\mathrm{and}g\left(z\right)=z-\frac{z^2}{2}.$$

The most important and extensively studied subclasses of $\mathcal{S}$ are the class ${\mathcal{S}}^{*}\left(ϵ\right)$ of star-like functions of order $ϵ$, and the class $\mathcal{K}\left(ϵ\right)$ of convex functions of order $ϵ$ in $\mathbb{D}$, which are defined by

$$\begin{array}{cc}\hfill & {\mathcal{S}}^{*}\left(ϵ\right):=\left\{f:f\in \mathcal{S}\mathrm{and}Re\left\{\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right\}>ϵ,(z\in \mathbb{D};0\le ϵ<1)\right\},\hfill \\ & \mathrm{and}\hfill \\ \hfill & \mathcal{K}\left(ϵ\right):=\left\{f:f\in \mathcal{S}\mathrm{and}Re\left\{1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\right\}>ϵ,(z\in \mathbb{D};0\le ϵ<1)\right\}.\hfill \end{array}$$

For $0\le ϵ<1$, a function $f\in \mathcal{F}$ is said to be in the class ${\mathcal{S}}_{\mathcal{F}}^{*}\left(ϵ\right)$ of bi-star-like functions of order $ϵ$ or the class ${\mathcal{K}}_{\mathcal{F}}\left(ϵ\right)$ of biconvex functions of order $ϵ$ if both f and $f^{-1}$ are, respectively, star-like or convex functions of order $ϵ$. The class we use in this paper is a linear combination of these two subclasses.

The study of the analytic and bi-univalent functions and the estimates on the first two coefficients, $|a_{{}_2}|$ $|a_{{}_3}|$, of various subclasses $\mathcal{F}$ is an active area of research in the complex analysis field. Lewin [11] studied the class $\mathcal{F}$ of bi-univalent functions and showed that $\left|a_{{}_2}\right|<1.51$. Thereafter, Brannan and Clunie [12] proposed that $\left|a_{{}_2}\right|<\sqrt{2}$. Netanyahu [13], as well, showed that $\underset{f\in \mathcal{F}}{max}\left|a_{{}_2}\right|=4/3$. For each $f\in \mathcal{F}$ given in $\left(5\right)$, obtaining the upper bounds on the Taylor–Maclaurin coefficients $|a_{{}_n}|(n\in N;n>3)$ is presumably still an open problem that has not been completely addressed. Kedzierawski [14] demonstrated the validity of the Brannan–Clunie conjecture for bi-star-like functions. Furthermore, Tan [15] found an upper bound for $|a_{{}_2}|$, which is the most accurate estimate for functions in the class $\mathcal{F}$, specifically, $|a_{{}_2}|<1.485$. Furthermore, Brannan and Taha [16] introduced the notions of strongly bi-star-like and bi-convex functions of $\beta$ ($0\le \beta <1$) and obtained estimates for the initial coefficients $|a_{{}_2}|$ and $|a_{{}_3}|$. Many other researchers have recently studied and discussed several subclasses of $\mathcal{F}$ and obtained coefficient bounds for $|a_{{}_2}|$ and $|a_{{}_3}|$.

In 1933, an inequality for the coefficients of univalent analytic functions was found by Fekete and Szegö [17]. They introduced the generalized functional $F^{\eta }\left(f\right)=a_{{}_3}-\eta a_{{}_2}^2$ where $0\le \eta <1$. The Fekete–Szegö inequality states that if $f\in \mathcal{F}$ is given by $\left(5\right)$, then

$$|F^{\eta }\left(f\right)|\le 1+2e^{\frac{-2\eta }{1-\eta }},$$

where $|F^{\eta }\left(f\right)|\le 1$ is obtained as $\eta \to 1^{-}$. Moreover, the coefficient functional $F^{\eta }\left(f\right)$ of the univalent analytic functions f of the $\left(5\right)$ plays a significant role in the field of geometric function theory. Note that the problem of maximizing $|F^{\eta }\left(f\right)|$ is called the the Fekete–Szegö problem. In many recent studies, researchers have obtained Fekete–Szegö inequalities for different classes of functions $f\in \mathcal{F}$ (see [18,19,20,21,22,23]).

In 2021, Ala Almourah et al. [24] introduced new upper bound estimations of the Taylor–Maclaurin coefficients and the Fekete–Szegö functional associated with certain subclasses of analytic and bi-univalent functions based on the classical GPs. In a recent published work [25], the characteristics of bi-univalent functions were investigated by introducing new subclasses subordinate to the q-GPs.

Recently, various subclasses of bi-univalent functions associated with GPs with its two reduced versions, $\alpha =\frac{1}{2}$ (LPs) and $\alpha =1$ (CPs of the second type), have been intensively studied by several researchers. All studies aim to determine more accurate estimations on the coefficients of these functions. Certain subclasses of the bi-univalent functions were identified based on specific properties and or conditions that enable more precise evaluations of their coefficients (see [26,27,28,29,30,31,32,33]). In this article, we consider a particular subclass of the bi-univalent functions subordinate to GPs to derive upper bounds for the Taylor–Maclaurin coefficients, $|a_2|$ and $|a_3|$, and determine the greatest value of the Fekete–Szegö functional $F^{\eta }\left(f\right)$.

2. Coefficient Bounds of the Class ${\Gamma }_{\mathcal{F}}^{\mathbf{\alpha }}(\mathbf{x},\mathbf{\lambda })$

Definition 1. 

Let $\lambda \in [0,1]$ and $x\in (\frac{1}{2},1]$. A function $f\in \mathcal{F}$ given by $\left(5\right)$ is said to be in the class ${\Gamma }_{\mathcal{F}}^{\alpha }(x,\lambda )$ with a non-zero real constant α if the following subordinations are satisfied

$$\lambda \left(1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\right)+(1-\lambda )\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)\prec H_{{}_{\alpha }}(x,z)$$

(7)

and

$$\lambda \left(1+\frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}\right)+(1-\lambda )\left(\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\right)\prec H_{{}_{\alpha }}(x,w),$$

(8)

where the function $g\left(w\right)=f^{-1}\left(w\right)$ is defined by $\left(6\right)$, and $H_{{}_{\alpha }}$ is the generating function of the GP given by $\left(2\right)$.

Remark 1 

([29]). For $\lambda =0$, we obtain the class ${\Gamma }_{\mathcal{F}}^{\alpha }(x,0)={\mathcal{S}}_{\mathcal{F}}^{*}(x,\alpha )$ that consists of function $f\in \mathcal{F}$ satisfying the conditions

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec H_{\alpha }(x,z)$$

(9)

and

$$\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\prec H_{\alpha }(x,w),$$

(10)

where the function $g=f^{-1}$ is defined by $\left(6\right)$.

Remark 2 

([29]). For $\lambda =1$, we obtain the class ${\Gamma }_{\mathcal{F}}^{\alpha }(x,1)={\mathcal{K}}_{\mathcal{F}}(x,\alpha )$ that consists of function $f\in \mathcal{F}$ satisfying the conditions

$$1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\prec H_{\alpha }(x,z)$$

(11)

and

$$1+\frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}\prec H_{\alpha }(x,w),$$

(12)

where the function $g=f^{-1}$ is defined by $\left(6\right)$.

Let $\mathsf{\Omega }$ be the class of all analytic functions $\omega \in \mathbb{D}$ which satisfy $\omega \left(0\right)=0$ and $\left|\omega \right(z\left)\right|<1$ for all $z\in \mathbb{D}$. We first recall the following lemma and then state the main result showing the coefficient estimates for the class ${\Gamma }_{\mathcal{F}}^{\alpha }(x,\lambda )$ given in Definition 1.

Lemma 1 

([2]). Let $\omega \in \mathsf{\Omega }$ with $\omega \left(z\right)={\sum }_{n=1}^{\infty }{\omega }_n{z}^n,z\in \mathbb{D}$. Then,

$$\left|{\omega }_1\right|\le 1,\left|{\omega }_n\right|\le 1-{\left|{\omega }_1\right|}^{2}forn\in \mathbb{N}\setminus \left\{1\right\}.$$

Theorem 1. 

Let $f\in \mathcal{F}$ of the form in (5) be in the class ${\Gamma }_{\mathcal{F}}^{\alpha }(x,\lambda )$. Then,

$$\left|a_2\right|\le \frac{{\left(2\left|\alpha \right|x\right)}^{\frac{3}{2}}}{\sqrt{(1+\lambda )\left[\left|2(1-\lambda ){\alpha }^2{x}^2+(1+\lambda )\alpha (1-2x^2)\right|+2(1+\lambda )\left|\alpha \right|x\right]}},$$

(13)

and

$$\left|a_3\right|\le \frac{{\left(2x\right)}^3{\left|\alpha \right|}^3}{(1+\lambda )\left[\left|2(1-\lambda ){\alpha }^2{x}^2+(1+\lambda )\alpha (1-2x^2)\right|+2(1+\lambda )\left|\alpha \right|x\right]}+\frac{x\left|\alpha \right|}{(1+2\lambda )}.$$

(14)

Proof. 

Let $f\in {\Gamma }_{\mathcal{F}}^{\alpha }(x,\lambda )$ for some $0\le \lambda \le 1$, and from $\left(7\right)$ and $\left(8\right)$ we have

$$\lambda \left(1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\right)+(1-\lambda )\left(\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\right)=H_{\alpha }(x,u\left(z\right))$$

(15)

and

$$\lambda \left(1+\frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}\right)+(1-\lambda )\left(\frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}\right)=H_{\alpha }(x,v\left(w\right)),$$

(16)

where $g\left(w\right)=f^{-1}\left(w\right)$ and $u,v\in \mathsf{\Omega }$ are given to be of the form

$$u\left(z\right)=\sum _{n=1}^{\infty }{c}_n{z}^{n}andv\left(w\right)=\sum _{n=1}^{\infty }{d}_n{w}^n.$$

From Lemma 1, we have

$$\left|c_n\right|\le 1\mathrm{and}\left|d_n\right|\le 1,n\in \mathbb{N}.$$

(17)

Then by taking $H_{\alpha }(x,z)$ given in (2), the right-hand sides of Equations (15) and (16) can be shown as follows

$$\begin{array}{cc}\hfill H_{\alpha }(x,u\left(z\right))& =1+C_1^{\left(\alpha \right)}\left(x\right)c_{1}z+\left[C_1^{\left(\alpha \right)}\left(x\right)c_2+C_2^{\left(\alpha \right)}\left(x\right)c_1^2\right]z^2\hfill \\ \hfill & +\left[C_1^{\left(\alpha \right)}\left(x\right)c_3+2C_2^{\left(\alpha \right)}\left(x\right)c_1{c}_2+C_3^{\left(\alpha \right)}\left(x\right)c_1^3\right]z^3+\cdots \hfill \end{array}$$

(18)

and

$$\begin{array}{cc}\hfill H_{\alpha }(x,v\left(w\right))& =1+C_1^{\left(\alpha \right)}\left(x\right)d_{1}w+\left[C_1^{\left(\alpha \right)}\left(x\right)d_2+C_2^{\left(\alpha \right)}\left(x\right)d_1^2\right]w^2\hfill \\ & +\left[C_1^{\left(\alpha \right)}\left(x\right)d_3+2C_2^{\left(\alpha \right)}\left(x\right)d_1{d}_2+C_3^{\left(\alpha \right)}\left(x\right)d_1^3\right]w^3+\cdots .\hfill \end{array}$$

(19)

Therefore, (15) and (16) become

$$\begin{array}{cc}\hfill \lambda & \left[1+2a_{2}z+(6a_3-4a_2^2)z^2+2(4a_2^3-9a_2{a}_3+6a_4)z^3+\cdots \right]\hfill \\ \hfill & +(1-\lambda )\left[1+a_{2}z+(2a_3-a_2^2)z^2+(a_2^3-3a_2{a}_3+3a_4)z^3+\cdots \right]\hfill \\ \hfill & =1+C_1^{\left(\alpha \right)}\left(x\right)c_{1}z+\left[C_1^{\left(\alpha \right)}\left(x\right)c_2+C_2^{\left(\alpha \right)}\left(x\right)c_1^2\right]z^2\hfill \\ & +\left[C_1^{\left(\alpha \right)}\left(x\right)c_3+2C_2^{\left(\alpha \right)}\left(x\right)c_1{c}_2+C_3^{\left(\alpha \right)}\left(x\right)c_1^3\right]z^3+\cdots ,\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill \lambda & \left[1-2a_{2}w+(8a_2^2-6a_3)w^2+(-32a_2^3+42a_2{a}_3-12a_4)w^3+\cdots \right]\hfill \\ \hfill & +(1-\lambda )\left[1-a_{2}w+(3a_2^2-2a_3)w^2+(-10a_2^3+12a_2{a}_3-3a_4)w^3+\cdots \right]\hfill \\ \hfill & =1+C_1^{\left(\alpha \right)}\left(x\right)d_{1}w+\left[C_1^{\left(\alpha \right)}\left(x\right)d_2+C_2^{\left(\alpha \right)}\left(x\right)d_1^2\right]w^2\hfill \\ & +\left[C_1^{\left(\alpha \right)}\left(x\right)d_3+2C_2^{\left(\alpha \right)}\left(x\right)d_1{d}_2+C_3^{\left(\alpha \right)}\left(x\right)d_1^3\right]w^3+\cdots .\hfill \end{array}$$

(21)

Now by equating the corresponding coefficients in (20) and (21), we obtain

$$(1+\lambda )a_2=C_1^{\left(\alpha \right)}\left(x\right)c_1,$$

(22)

$$2(1+2\lambda )a_3-(1+3\lambda )a_2^2=C_1^{\left(\alpha \right)}\left(x\right)c_2+C_2^{\left(\alpha \right)}\left(x\right)c_1^2,$$

(23)

$$-(1+\lambda )a_2=C_1^{\left(\alpha \right)}\left(x\right)d_1,$$

(24)

and

$$(3+5\lambda )a_2^2-2(1+2\lambda )a_3=C_1^{\left(\alpha \right)}\left(x\right)d_2+C_2^{\left(\alpha \right)}\left(x\right)d_1^2.$$

(25)

From (22) and (24), we obtain the following two equations

$$c_1=-d_1$$

(26)

and

$$c_1^2+d_1^2=\frac{2{(1+\lambda )}^2{a}_2^2}{{\left[C_1^{\left(\alpha \right)}\left(x\right)\right]}^2},$$

(27)

and from (23), (25) and (27), we obtain

$$a_2^2=\frac{{\left(C_1^{\left(\alpha \right)}\left(x\right)\right)}^3(c_2+d_2)}{2(1+\lambda )\left[{\left(C_1^{\left(\alpha \right)}\left(x\right)\right)}^2-(1+\lambda )C_2^{\left(\alpha \right)}\left(x\right)\right]}.$$

(28)

By applying Lemma 1, and considering Equations (22) and (26), one can obtain

$$\begin{array}{cc}\hfill {\left|a_2\right|}^2& \le \frac{{\left|C_1^{\left(\alpha \right)}\left(x\right)\right|}^3\left(1-{\left|c_1\right|}^2\right)}{(1+\lambda )\left|{\left(C_1^{\left(\alpha \right)}\left(x\right)\right)}^2-(1+\lambda )C_2^{\left(\alpha \right)}\left(x\right)\right|}\hfill \\ \hfill & =\frac{{\left|C_1^{\left(\alpha \right)}\left(x\right)\right|}^3}{(1+\lambda )\left[\left|{\left(C_1^{\left(\alpha \right)}\left(x\right)\right)}^2-(1+\lambda )C_2^{\left(\alpha \right)}\left(x\right)\right|+(1+\lambda )\left|C_1^{\left(\alpha \right)}\left(x\right)\right|\right]},\hfill \end{array}$$

(29)

and therefore

$$\left|a_2\right|\le \frac{\left|C_1^{\left(\alpha \right)}\left(x\right)\right|\sqrt{\left|C_1^{\left(\alpha \right)}\left(x\right)\right|}}{\sqrt{(1+\lambda )\left[\left|{\left(C_1^{\left(\alpha \right)}\left(x\right)\right)}^2-(1+\lambda )C_2^{\left(\alpha \right)}\left(x\right)\right|+(1+\lambda )\left|C_1^{\left(\alpha \right)}\left(x\right)\right|\right]}}.$$

(30)

Substituting $C_1^{\left(\alpha \right)}\left(x\right)$ and $C_2^{\left(\alpha \right)}\left(x\right)$, as given in (1), into Equation (30) yields

$$\left|a_2\right|\le \frac{{\left(2\left|\alpha \right|x\right)}^{\frac{3}{2}}}{\sqrt{(1+\lambda )\left[\left|2(1-\lambda ){\alpha }^2{x}^2+(1+\lambda )\alpha (1-2x^2)\right|+2(1+\lambda )\left|\alpha \right|x\right]}}.$$

To obtain an estimation of $\left|a_3\right|$, we first subtract Equation (25) from Equation (23),

$$a_3=a_2^2+\frac{C_1^{\left(\alpha \right)}\left(x\right)(c_2-d_2)}{4(1+2\lambda )},$$

(31)

and then obtain

$$\left|a_3\right|\le {\left|a_2\right|}^2+\frac{\left|C_1^{\left(\alpha \right)}\left(x\right)\right|\left|c_2-d_2\right|}{4(1+2\lambda )}.$$

(32)

By applying Lemma 1 and using Equation (1), we have

$$\left|a_3\right|\le \frac{{\left(2x\right)}^3{\left|\alpha \right|}^3}{(1+\lambda )\left[\left|2(1-\lambda ){\alpha }^2{x}^2+(1+\lambda )\alpha (1-2x^2)\right|+2(1+\lambda )\left|\alpha \right|x\right]}+\frac{x\left|\alpha \right|}{(1+2\lambda )}.$$

(33)

This completes the proof of Theorem 1. □

In the following section, utilizing the values of $a_2^2$ and $a_3$ helps to establish the Fekete–Szegö inequality for functions $f\in {\Gamma }_{\mathcal{F}}^{\alpha }(x,\lambda )$.

3. Fekete–Szegö Functional Estimations of the Class ${\Gamma }_{\mathcal{F}}^{\mathbf{\alpha }}(\mathbf{x},\mathbf{\lambda })$

Theorem 2. 

Let $f\in \mathcal{F}$ given by the form (5) be in the class ${\Gamma }_{\mathcal{F}}^{\alpha }(x,\lambda )$. Then

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}\frac{2x\left|\alpha \right|}{1+2\lambda }\hfill & if0\le \left|h\left(\eta \right)\right|\le \frac{1}{4(1+2\lambda )}\hfill \\ 8x\left|\alpha \right|\left|h\left(\eta \right)\right|\hfill & if\left|h\left(\eta \right)\right|\ge \frac{1}{4(1+2\lambda )}.\hfill \end{array}\right.$$

Proof. 

From Equations (28) and (31), we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill a_3-\eta a_2^2& =a_2^2+\frac{C_1^{\left(\alpha \right)}\left(x\right)(c_2-d_2)}{4(1+2\lambda )}-\eta a_2^2\hfill \\ \hfill & =(1-\eta )a_2^2+\frac{C_1^{\left(\alpha \right)}\left(x\right)(c_2-d_2)}{4(1+2\lambda )}\hfill \\ \hfill & =(1-\eta )\frac{{\left(C_1^{\left(\alpha \right)}\left(x\right)\right)}^3(c_2+d_2)}{2(1+\lambda )\left[{\left(C_1^{\left(\alpha \right)}\left(x\right)\right)}^2-(1+\lambda )C_2^{\left(\alpha \right)}\left(x\right)\right]}+\frac{C_1^{\left(\alpha \right)}\left(x\right)(c_2-d_2)}{4(1+2\lambda )}\hfill \\ \hfill & =2\alpha x\left(\left[h\left(\eta \right)+\frac{1}{4(1+2\lambda )}\right]c_2+\left[h\left(\eta \right)-\frac{1}{4(1+2\lambda )}\right]d_2\right),\hfill \end{array}\end{array}$$

where

$$h\left(\eta \right)=\frac{2\alpha x^2(1-\eta )}{(1+\lambda )\left[4\alpha x^2-(1+\lambda )(2(1+\alpha )x^2-1)\right]}.$$

Then, in view of (1), and using $\left(17\right)$ we conclude that

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}\frac{2x\left|\alpha \right|}{1+2\lambda }\hfill & \mathrm{if}0\le \left|h\left(\eta \right)\right|\le \frac{1}{4(1+2\lambda )}\hfill \\ 8x\left|\alpha \right|\left|h\left(\eta \right)\right|\hfill & \mathrm{if}\left|h\left(\eta \right)\right|\ge \frac{1}{4(1+2\lambda )}.\hfill \end{array}\right.$$

This completes the proof of Theorem 2. □

Corollary 1. 

Let $f\in \mathcal{F}$ given by the form (5) be in the class ${\mathcal{S}}_{\mathcal{F}}^{*}(x,\alpha )$. Then

$$\left|a_2\right|\le \frac{{\left(2\left|\alpha \right|x\right)}^{\frac{3}{2}}}{\sqrt{\left|2{\alpha }^2{x}^2+\alpha (1-2x^2)\right|+2\left|\alpha \right|x}},$$

$$\left|a_3\right|\le \frac{{\left(2x\right)}^3{\left|\alpha \right|}^3}{\left|2{\alpha }^2{x}^2+\alpha (1-2x^2)\right|+2\left|\alpha \right|x}+x\left|\alpha \right|,$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}2x\left|\alpha \right|\hfill & if0\le \left|h_1\left(\eta \right)\right|\le \frac{1}{4};\hfill \\ 8x\left|\alpha \right|\left|h_1\left(\eta \right)\right|\hfill & if\left|h_1\left(\eta \right)\right|\ge \frac{1}{4},\hfill \end{array}\right.$$

where

$$h_1\left(\eta \right)=\frac{2\alpha x^2(1-\eta )}{2\alpha x^2+(1-2x^2)}.$$

Corollary 2. 

Let $f\in \mathcal{F}$ given by the form (5) be in the class ${\mathcal{K}}_{\mathcal{F}}(x,\alpha )$. Then

$$\left|a_2\right|\le \frac{{\left(\left|\alpha \right|x\right)}^{\frac{3}{2}}}{\sqrt{\left|\frac{1}{2}\alpha (1-2x^2)\right|+\left|\alpha \right|x}},$$

$$\left|a_3\right|\le \frac{{\left(\left|\alpha \right|x\right)}^3}{\sqrt{\left|\frac{1}{2}\alpha (1-2x^2)\right|+\left|\alpha \right|x}}+\frac{x\left|\alpha \right|}{3},$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}\frac{2x\left|\alpha \right|}{3}\hfill & if0\le \left|h_2\left(\eta \right)\right|\le \frac{1}{12};\hfill \\ 8x\left|\alpha \right|\left|h_2\left(\eta \right)\right|\hfill & if\left|h_2\left(\eta \right)\right|\ge \frac{1}{12},\hfill \end{array}\right.$$

where

$$h_2\left(\eta \right)=\frac{\alpha x^2(1-\eta )}{1-4x^2}.$$

4. Conclusions

In our present work, new upper bound estimations of the Taylor–Maclaurin coefficients $|a_2|$ and $|a_3|$, and the Fekete–Szegö functional $F^{\eta }\left(f\right)=a_{{}_3}-\eta a_{{}_2}^2$ were derived using a certain subclass of the normalized analytic and bi-univalent functions on the open unit disk $\mathbb{D}$ associated with orthogonal GPs satisfying the subordination conditions on $\mathbb{D}$. For future research, the upper bound estimations and inequalities for the second Hankel determinant of functions belonging to this univalent function subclass will be investigated. Furthermore, we aim to construct a new subclass of analytic bi-univalent functions defined on the symmetric domain by means of GPs with distribution series to estimate the upper bound of the Taylor–Maclaurin coefficients and the Fekete–Szegö functional.