Applications of Orthogonal Polynomials to Subclasses of Bi-Univalent Functions

Abstract

Orthogonal polynomials have been studied extensively by Legendre in 1784. They are representatively related to typically real functions, which played an important role in the geometric function theory, and its role of estimating coefficient bounds. This chapter associates certain bi-univalent functions with certain orthogonal polynomials, such as Gegnbauer polynomials and Horadam polynomials, and then explores some properties of the subclasses in hand. This chapter is concerned with the connection between orthogonal polynomials and bi-univalent functions. Our purpose is to introduce certain classes of bi-univalent functions by means of Gegenbauer polynomials and Hordam polynomials. Bounds for the initial coefficients of $\mid a_2\mid$ and $\mid a_3\mid$, and results related to Fekete–Szegö functional are obtained.

1. Introduction

Orthogonal polynomials appear in many areas of mathematics and play a vital role in the development of numerical and analytical approaches. Also, many mathematicians have been interested in its subjects. Orthogonal polynomials are gaining traction in current different mathematics, such as operator theory, number theory, special functions, analytic functions, and approximation theory. Also, has a wide range applications in physics and engineering fields. The subject of orthogonal polynomials finds its origins in 1784 by Legendre [1]. In the 18th century, the first examples of orthogonal polynomials were developed by brilliant mathematicians, before the general theory, which appeared in the 19th century. Orthogonal polynomials have been found to have connections with trigonometric, hypergeometric, Bessel, and elliptic functions, helping to solve certain problems in the theory of differential and integral equations, and in quantum mechanics and mathematical statistics. Up until the late 20th century, Szegö [2] covered most of the general theories along with all standard formulas for the three classical orthogonal polynomials. The connection of orthogonal polynomials with other branches of mathematics is very deep and impressive.

Officially, the classes of polynomials $P_n$ defined over a range $\left[a,b\right]$ are orthogonal satisfy

$${\mathit{\text{degP}}}_n=n,n=\mathrm{0,1,2,3},\dots$$

and

$$\underset{a}{\overset{b}{\int }}{P}_n{P}_{m}W\left(x\right)\mathit{dx}=0,m\ne n,$$

where $W\left(x\right)$ is nonnegative function in (a, b).

Orthogonal polynomials are among the most often studied polynomials, such as Gegenbauer polynomials, Hordam polynomials, Chebyshev polynomials of the first and the second kind, Laguerre polynomials, and Jacobi polynomial. Recently, several papers from a theoretical point of view and in the case of bi-univalent functions have been studied.

The main goal of this chapter focuses on some original results of bi-univalent functions by using Gegenbauer polynomials and Hordam polynomials. Estimates on the initial Taylor-Maclaurin coefficients and the Fekete-Szegö inequalities for some subclasses of bi univalent functions are obtained. Also, we give several illustrative examples of the bi-univalent function subclass, which we introduce here. To do so, we take into account the following definitions.

Let $\mathcal{A}$ represents the class of all functions of the form

$$f\left(z\right)=z+\sum _{n=2}^{\infty }{a}_n{z}^n,$$E1

which are analytic in the open unit open disk $U=\left\{z:z\in C,|z|<1\right\}$, and let $\mathcal{S}$ be the class of all functions in $\mathcal{A}$_, which are univalent and normalized by the conditions

$$f\left(0\right)=0=f^{\prime }\left(0\right)-1$$

in $U$.

For any two analytic function $f_1$ and $f_2$ in unit disk $U$, we say that $f_1$ is subordinate to $f_2$, and denoted by $f_1\prec f_2$, if there exists Schwarz function

$$\varpi \left(z\right)=\sum _{n=1}^{\infty }{\mathbf{c}}_n{z}^n\left(\varpi \left(0\right)=0,\left|\varpi \left(z\right)\right|<1\right),$$E2

analytic in $U$ such that

$$f_1\left(z\right)=f_2\left(\varpi \left(z\right)\right)\left(z\in U\right),$$E3

where $\left|{\mathbf{c}}_n\right|\le 1$ (see [3] for $\varpi \left(z\right)$).

In particular, it is known that

$$f\left(z\right)\prec g\left(z\right)\left(z\in U\right)\iff f\left(0\right)=g\left(0\right)\mathit{\text{and}}f\left(U\right)\subset g\left(U\right).$$

Thus, clearly, every univalent function $f$ has an inverse $f^{-1}$, defined by

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

and

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

where

$$f^{-1}\left(w\right)=w+a_2{w}^2+\left(2a_2^2-3a_3\right)w^3-\left(5a_2^3-5a_2{a}_3+a_4\right)w^4+\cdots .$$E4

If $f$ and $f^{-1}$ are univalent in $U$, then a function $f\in \mathcal{A}$ is called bi-univalent.

The study of the class $\mathrm{\Sigma }$ of bi-univalent functions was discussed by Lewin [4] while Brannan and Taha [5] derived estimates for the initial coefficients. Lately, Srivastava et al. [6] have actually revived the investigation of analytic and bi-univalent functions. Several researchers have investigated and examined various subclasses of analytic and bi-univalent functions, one can refer to the works of [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20].

Ma and Minda [21] investigated the class of starlike and convex functions as the following

$$S^{\ast }\left(\varphi \right)=\left\{f:f\in \mathcal{A},\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec \varphi \left(z\right)\right\},z\in U,$$

and

$$C\left(\varphi \right)=\left\{f:f\in \mathcal{A},1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\prec \varphi \left(z\right)\right\},z\in U.$$

Clear that, if $f\left(z\right)\in C\left(\varphi \right)$, then $z{f}^{\prime }\left(z\right)\in S^{\ast }\left(\varphi \right)$.

Initiating an investigation on properties of bi-univalent functions linked by Gegenbauer polynomials and Hordam polynomials will be discussed in the following sections.

1.1 Applications of Gegenbauer polynomials to subclasses of bi-univalent functions

This section is devoted to studying and discussing Gegenbauer Polynomials by means of bi-univalent function. We first introduce the following definitions.

A generating function of Gegenbauer polynomials of the sequence $C_n^{\alpha }\left(x\right),n\in \mathbb{N}$ is defined by the following:

$$H_{\gamma }\left(x,z\right)=\sum _{n=0}^{\infty }{C}_n^{\alpha }\left(x\right)z^n=\frac{1}{{\left(1-2\mathit{xz}+z^2\right)}^{\alpha }},$$E5

where $\alpha$ is nonzero real constant, $x\in \left[-1,1\right]$, $z\in \mathbb{U}$_, and $C_n^{\alpha }\left(x\right)$ is defined by

$$C_n^{\alpha }\left(x\right)=\frac{2x\left(n+\alpha -1\right)C_{n-1}^{\alpha }\left(x\right)-\left(n+2\alpha -2\right)C_{n-2}^{\alpha }\left(x\right)}{n}.$$E6

It is clear that, $C_0^{\alpha }\left(x\right)=1,C_1^{\alpha }\left(x\right)=2\mathit{\alpha x}$ and $C_2^{\alpha }\left(x\right)=2\alpha \left(1+\alpha \right)x^2-\alpha .$ Furthermore, we present some particular cases of $C_n^{\alpha }\left(x\right)$ as follows:

1. For $\alpha =0$, we get the Chebyshev polynomials.

2. For $\alpha =\frac{1}{2}$, we get the Legendre polynomials.

A class of starlike bi-univalent functions is linked with Gegenbauer polynomial as follows.

Definition 2.1. Let $\gamma \in \left(-1/2,\infty \right)\backslash \left\{0\right\}$, and $0\le \lambda \le 1,$ $x\in \left(1/2,1\right).$ A function $f\in \mathrm{\Sigma }$ given by Eq. (1) is said to be in the class ${\mathcal{Q}}_{{\mathrm{\Sigma }}_s}\left(H_{\gamma }\right)$ if there exist the following functions:

$$g\left(z\right)=z+\sum _{n=2}^{\infty }{b}_n{z}^n\in {\mathcal{S}}^{\ast }\left(1/2\right),G\left(w\right)=w+\sum _{n=2}^{\infty }{t}_n{w}^n\in {\mathcal{S}}^{\ast }\left(1/2\right)$$

and the conditions are fulfilled

$$\frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}\prec H_{\gamma }\left(x,z\right)\left(z\in \mathbb{U}\right),$$E7

and

$$\frac{w{F}^{\prime }\left(w\right)}{F\left(w\right)}\prec H_{\gamma }\left(x,w\right)\left(w\in \mathbb{U}\right),$$E8

where $F$ is the inverse of $f$ is defined by Eq. (4) and $H_{\gamma }$ is the generating function of the Gegenbauer polynomial given by Eq. (5).

Also, a class of convex bi-univalent functions linked with Gegenbauer polynomial as follows:

Definition 2.2. Let $\gamma \in \left(-1/2,\infty \right)\backslash \left\{0\right\}$ and $0\le \lambda \le 1,$ $x\in \left(1/2,1\right).$ A function $f\in \mathrm{\Sigma }$ given by $\left(??\right)$ is said to be in the class ${{\mathcal{Q}}^{*}}_{{\mathrm{\Sigma }}_s}\left(H_{\gamma }\right)$ if there exist the following functions:

$$g\left(z\right)=z+\sum _{n=2}^{\infty }{b}_n{z}^n\in {\mathcal{S}}^{\ast }\left(1/2\right),G\left(w\right)=w+\sum _{n=2}^{\infty }{t}_n{w}^n\in {\mathcal{S}}^{\ast }\left(1/2\right)$$

and the conditions are fulfilled

$$1+\frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}\prec H_{\gamma }\left(x,z\right)\left(z\in \mathbb{U}\right),$$E9

and

$$\frac{w{F}^{″}\left(w\right)}{F^{\prime }\left(w\right)}\prec H_{\gamma }\left(x,w\right)\left(w\in \mathbb{U}\right),$$E10

where $F$ is the inverse of $f$ is defined by Eq. (4) and $H_{\gamma }$ is the generating function of the Gegenbauer polynomial given by Eq. (5).

Next section, we discuss some recent results of bi-univalent functions for several families associated with Gegenbauer polynomials and Hordam polynomials, respectively. For the proofs and details of the main theorems, one can refer to [7, 8, 9], respectively.

1.2 Initial Taylor coefficient estimates for the functions of ${\mathcal{Q}}_{{\mathrm{\Sigma }}_s}\left(H_{\gamma }\right)$ and ${{\mathcal{Q}}^{*}}_{{\mathrm{\Sigma }}_s}\left(H_{\gamma }\right)$

We begin this section by defining Fekete-Szegö inequality, which was given by Fekete and Szegö [2] and defined as follows.

If $f\in S$ and $\eta$ is real, then

$$\left|a_3-\eta a_2^2\right|\le 1+2e^{\frac{-2\eta }{1-\mu }}.$$E11

This bound is sharp.

Theorem 2.1. Let the function $f$ given by Eq. (1) be in the class ${\mathcal{Q}}_{{\mathrm{\Sigma }}_s}\left(H_{\gamma }\right)$. Then

$$\left|a_2\right|\le \frac{2\left|\gamma \right|x\sqrt{2\left|\gamma \right|x}}{\sqrt{\mid 4\gamma \left(\gamma -1\right)x^2+2\gamma \mid }}$$E12

and

$$\left|a_3\right|\le {\gamma }^2{x}^2+\frac{\left|\gamma \right|x}{3}.$$E13

For some $\eta \in \mathbb{R}$, we have

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{l}\frac{\left|\gamma \right|x}{3},\mathit{\text{if}}\left|\eta -1\right|\le \left|\frac{1-2x^2}{6\gamma x^2}\right|\\ \\ \frac{2{\gamma }^2{x}^3\left|1-\eta \right|}{\left|1-2x^2\right|},\mathit{\text{if}}\left|\eta -1\right|\le \left|\frac{1-2x^2}{6\gamma x^2}\right|\end{array}\right.$$E14

For $\gamma =1$ in Theorem 2.1, we have the following corollary.

Corollary 2.1. Let the function $f$ given by Eq. (1) be in the class ${\mathcal{Q}}_{{\mathrm{\Sigma }}_s}\left(H\right)$. Then

$$\left|a_2\right|\le 2x\sqrt{x}$$E15

and

$$\left|a_3\right|\le x^2+\frac{x}{3}.$$E16

For some $\eta \in \mathbb{R}$, we have

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{l}\frac{x}{3},\mathit{\text{if}}\left|\eta -1\right|\le \left|\frac{1-2x^2}{6x^2}\right|\\ \\ \frac{2x^3\left|1-\eta \right|}{\left|1-2x^2\right|},\mathit{\text{if}}\left|\eta -1\right|\le \left|\frac{1-2x^2}{6x^2}\right|\end{array}\right.$$E17

Theorem 2.2. Let the function $f$ given by Eq. (1) be in the class ${{\mathcal{Q}}^{*}}_{{\mathrm{\Sigma }}_s}\left(H_{\gamma }\right)$. Then

$$\left|a_2\right|\le \frac{2\left|\gamma \right|x\sqrt{2\left|\gamma \right|x}}{\sqrt{\mid 2\gamma \left(\gamma -1\right)x^2+\gamma \mid }}$$E18

and

$$\left|a_3\right|\le 4{\gamma }^2{x}^2+\left|\gamma \right|x.$$E19

For some $\eta \in \mathbb{R}$, we have

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{l}\left|\gamma \right|x,\mathit{\text{if}}\left|\eta -1\right|\le \left|\frac{2\gamma x^2-2x^2+1}{2\gamma x^2}\right|\\ \\ \frac{8{\left|\gamma \right|}^3{x}^3\left|1-\eta \right|}{\left|2\gamma \left(\gamma -1\right)x^2+\gamma \right|},\mathit{\text{if}}\left|\eta -1\right|\le \left|\frac{2\gamma x^2-2x^2+1}{2\gamma x^2}\right|\end{array}\right.$$E20

For $\gamma =1$ in Theorem 2.2, we have the following corollary.

Corollary 2.2. Let the function $f$ given by Eq. (1) be in the class ${{\mathcal{Q}}^{*}}_{{\mathrm{\Sigma }}_s}\left(H_{\gamma }\right)$. Then

$$\left|a_2\right|\le 2x\sqrt{2x}$$E21

and

$$\left|a_3\right|\le 4x^2+x.$$E22

For some $\eta \in \mathbb{R}$, we have

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{l}x,\mathit{\text{if}}\left|\eta -1\right|\le \frac{1}{2x^2}\\ \\ 8x^3\left|1-\eta \right|,\mathit{\text{if}}\left|\eta -1\right|\le \frac{1}{2x^2}\end{array}\right.$$E23

Next section is devoted to studying and discussing Hordam polynomials by means of bi-univalent function.

1.3 Applications of Hordam polynomials to subclasses of bi-univalent functions

We begin this section by introducing the recurrence relation of the Hordam polynomials, which was studied by Horzum and Koçer [22] as follows.

$$h_n\left(x\right)={\mathit{pxh}}_{n-1}\left(x\right)+{\mathit{qh}}_{n-2}\left(x\right);\left(n\in \mathbb{N}\ge 2\right),$$E24

with

$$h_1\left(x\right)=a,h_2\left(x\right)=\mathit{bx},h_3\left(x\right)={\mathit{pbx}}^2+\mathit{pq},$$E25

where $a,b,p$_, and $q$ are some real constants, and the characteristic equation of above recurrence relation is

$$t^2-\mathit{pxt}-q=0,$$E26

with two real roots:

$$\alpha =\frac{\mathit{px}+\sqrt{p^2{x}^2+4q}}{2},$$

and

$$\beta =\frac{\mathit{px}-\sqrt{p^2{x}^2+4q}}{2}.$$

Selecting particular values of $a,b,p$_, and $q$ reduces to special various polynomials as follows:

• If $a=b=p=q=1$, the Fibonacci polynomials sequence is obtained

  $$F_n\left(x\right)={\mathit{xF}}_{n-1}\left(x\right)+F_{n-2}\left(x\right),F_1\left(x\right)=1,F_2\left(x\right)=x.$$

• If $a=2,b=p=q=1$, the Lucas polynomials sequence is acquired

  $$L_{n-1}\left(x\right)={\mathit{xL}}_{n-2}\left(x\right)+L_{n-3}\left(x\right),L_0\left(x\right)=2,L_1\left(x\right)=x.$$

• If $a=q=1,b=p=2$, the Pell polynomials sequence is attained

  $$P_n\left(x\right)=2{\mathit{xP}}_{n-1}\left(x\right)+P_{n-2}\left(x\right),p_1\left(x\right)=1,P_2\left(x\right)=2x.$$

• If $a=b=p=2,q=1$, the Pell-Lucas polynomials sequence is obtained

  $$Q_{n-1}\left(x\right)=2{\mathit{xQ}}_{n-2}\left(x\right)+Q_{n-3}\left(x\right),Q_0\left(x\right)=2,Q_1\left(x\right)=2x.$$

• If $a=1,b=p=2,q=-1$, the Chebyshev polynomials of second kind sequence are acquired

  $$U_{n-1}\left(x\right)=2{\mathit{xU}}_{n-2}\left(x\right)+U_{n-3}\left(x\right),U_0\left(x\right)=1,U_1\left(x\right)=2x.$$

• If $a=b=1,p=2,q=-1$, the Chebyshev polynomials of first kind sequence are obtained

  $$T_{n-1}\left(x\right)=2{\mathit{xT}}_{n-2}\left(x\right)+T_{n-3}\left(x\right),T_0\left(x\right)=1,T_1\left(x\right)=x.$$

• If $x=1$, the Horadam numbers sequence is derived

  $$h_{n-1}\left(1\right)={\mathit{ph}}_{n-2}\left(1\right)+{\mathit{qh}}_{n-3}\left(1\right),h_0\left(1\right)=a,h_1\left(1\right)=b.$$

For more details related to these polynomial sequences succession, can refer to [22, 23, 24, 25].

The generating function of the Horadam polynomials $h_n\left(x\right)$ is studied by Horadam [23] and defined as follows:

$$\mathrm{\Omega }\left(x,z\right)=\frac{a+\left(b-\mathit{ap}\right)\mathit{xt}}{1-\mathit{pxt}-{\mathit{qt}}^2}=\sum _{n=1}^{\infty }{h}_n\left(x\right)z^{n-1}.$$E27

New subclasses of the bi-univalent function class $\mathrm{\Sigma }$ associated with Horadam polynomial are presented in the following.

Definition 3.1. A function $f\in {\mathrm{\Sigma }}^{\prime }$ given by Eq. (1) is said to be in the class ${\mathrm{\Sigma }}^{\prime }\left(x\right)$ if the following subordinations hold:

$$f^{\prime }\left(z\right)\prec \mathrm{\Omega }\left(x,z\right)+1-\alpha$$E28

and

$$g^{\prime }\left(w\right)\prec \mathrm{\Omega }\left(x,w\right)+1-\alpha ,$$E29

where the real constants $a$, $b$, and $q$ are as in Eq. (25) and $g=f^{-1}$ is given by Eq. (4).

Theorem 3.1. Let the function $f\in {\mathrm{\Sigma }}^{\prime }$ given by Eq. (1) be in the class ${\mathrm{\Sigma }}^{\prime }\left(x\right)$. Then

$$\mid a_2\mid \le \frac{\mid \mathit{bx}\mid \sqrt{\mid \mathit{bx}\mid }}{\sqrt{\left|{\mathit{bx}}^2\left(3b-4p\right)-4\mathit{aq}\right|}}$$E30

$$\mid a_3\mid \le \frac{\mid \mathit{bx}\mid }{3}+\frac{{\left(\mathit{bx}\right)}^2}{4},$$E31

and for some $\eta \in R$,

$$\mid a_3-\eta a_2^2\mid \le \left\{\begin{array}{c}\frac{\mid 2\mathit{bx}\mid }{3},\mid \eta -1\mid \le 1-\frac{\mid 4\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }{3b^2{x}^2}\\ \frac{{\left|\mathit{bx}\right|}^3\mid 1-\eta \mid }{3b^2{x}^2-4\left({\mathit{pbx}}^2+\mathit{qa}\right)},\mid \eta -1\mid \ge 1-\frac{\mid 4\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }{3b^2{x}^2}\end{array}\right\}.$$E32

In light of relation (27), Theorem 3.1, we can readily deduce the following corollaries.

Corollary 3.1. For $t\in \left(1/2,1\right)$, let the function $f\in {\mathrm{\Sigma }}^{\prime }$ given by Eq. (1) be in the class ${\mathrm{\Sigma }}^{\prime }\left(t\right)$. Then

$$\mid a_2\mid \le \frac{t\sqrt{2t}}{\sqrt{\left|1-t^2\right|}}$$E33

$$\mid a_3\mid \le \frac{2t}{3}+t^2,$$E34

and for some $\eta \in R$,

$$\mid a_3-\eta a_2^2\mid \le \begin{array}{c}\left\{\begin{array}{c}\frac{4t}{3},\mid \eta -1\mid \le \frac{1-t^2}{3t^2}\\ \frac{2\mid \eta -1\mid }{1-t^2},\mid \eta -1\mid \ge \frac{1-t^2}{3t^2}\end{array}\right\}.\end{array}$$E35

Taking $\eta =1$ in Corollary 3.1, we get the following corollary.

Corollary 3.2. For $t\in \left(1/2,1\right)$, let the function $f\in {\mathrm{\Sigma }}^{\prime }$ given by Eq. (1) be in the class ${\mathrm{\Sigma }}^{\prime }\left(t\right)$. Then

$$\mid a_3-a_2^2\mid \le \frac{4t}{3}.$$E36

The class $S_s^{\ast }$ of functions starlike with respect to symmetric points is introduced by Sakaguchi [26], consisting of functions $f\in S$ that satisfy the following condition

$$\Re \left(\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f\left(-z\right)}\right)>0,z\in \mathbb{U}.$$

Similarly, the class $K_s$ of functions convex with respect to symmetric points is introduced by Wang et al. [27], consisting of functions $f\in S$ that satisfy the following condition

$$\Re \left(\frac{2{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{f^{\prime }\left(z\right)+f^{\prime }\left(-z\right)}\right)>0,z\in \mathbb{U}.$$

Moreover, Ravichandran [28] introduced the following two subclasses:

For such a function $\varphi$, then a function $f\in A$ is in the class $S_s^{\ast }\left(\varphi \right)$ if

$$\frac{2z{f}^{\prime }\left(z\right)}{f\left(z\right)-f\left(-z\right)}\prec \varphi \left(z\right),z\in \mathbb{U},$$

and in the class $K_s\left(\varphi \right)$ if

$$\frac{2{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }}{f^{\prime }\left(z\right)+f^{\prime }\left(-z\right)}\prec \varphi \left(z\right)z\in \mathbb{U}.$$

Recently, a new class $L_{\lambda }$ of $\lambda$-pseudo-starlike functions is defined by Babalola [29] as the following:

Let $f\in A$ and $\lambda \ge 1$ is real. Then $f\left(z\right)\in L_{\lambda }$ of $\lambda$-pseudo-starlike functions in $\mathbb{U}$ if and only if

$$\Re \left\{\frac{z{\left(f^{\prime }\left(z\right)\right)}^{\lambda }}{f\left(z\right)}\right\}\ge 0,z\in \mathbb{U}.$$

More recently, the author introduced and studied two subclasses $L\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$ and $M\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$ of $\lambda$-pseudo-bi-univalent functions with respect to symmetrical points linked by the Horadam polynomials $h_n\left(x\right)$ and the generating function $\mathrm{\Omega }\left(x,z\right)$ as follows.

Definition 3.2. A function $f\in \mathrm{\Sigma }$ given by Eq. (1) is said to be in the class $L\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$ if the following conditions are satisfied:

$$\left(1-\alpha \right)\frac{2z{\left[f^{\prime }\left(z\right)\right]}^{\lambda }}{f\left(z\right)-f\left(-z\right)}+\alpha \frac{2{\left[{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }\right]}^{\lambda }}{{\left[f\left(z\right)-f\left(-z\right)\right]}^{\prime }}\prec \mathrm{\Omega }\left(x,z\right)+1-\alpha$$E37

and

$$\left(1-\alpha \right)\frac{2w{\left[g^{\prime }\left(w\right)\right]}^{\lambda }}{g\left(w\right)-g\left(-w\right)}+\alpha \frac{2{\left[{\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\lambda }}{{\left[g\left(w\right)-g\left(-w\right)\right]}^{\prime }}\prec \mathrm{\Omega }\left(x,w\right)+1-\alpha$$E38

where the real constants $a$, $b$, and $q$ are as in Eq. (25) and $g\left(w\right)=f^{-1}\left(z\right)$ is given by Eq. (4).

Definition 3.3. A function $f\in \mathrm{\Sigma }$ given by Eq. (1) is said to be in the class $M\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$ if the following conditions are satisfied:

$${\left(\frac{2z{\left[f^{\prime }\left(z\right)\right]}^{\lambda }}{f\left(z\right)-f\left(-z\right)}\right)}^{\alpha }{\left(\frac{2{\left[{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }\right]}^{\lambda }}{{\left[f\left(z\right)-f\left(-z\right)\right]}^{\prime }}\right)}^{1-\alpha }\prec \mathrm{\Omega }\left(x,z\right)+1-\alpha$$E39

and

$${\left(\frac{2w{\left[g^{\prime }\left(w\right)\right]}^{\lambda }}{g\left(w\right)-g\left(-w\right)}\right)}^{\alpha }{\left(\frac{2{\left[{\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\lambda }}{{\left[g\left(w\right)-g\left(-w\right)\right]}^{\prime }}\right)}^{1-\alpha }\prec \mathrm{\Omega }\left(x,w\right)+1-\alpha$$E40

where the real constants $a$, $b$, and $q$ are as in Eq. (25) and $g\left(w\right)=f^{-1}\left(z\right)$ is given by Eq. (4).

In the next theorems, the coefficient bounds and Fekete-Szegö type inequalities for the function subclasses $L\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$ and $M\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$, respectively.

Theorem 3.2. Let the function $f\in \mathrm{\Sigma }$ given by Eq. (1) be in the class $L\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$. Then

$$\mid a_2\mid \le \frac{\mid \mathit{bx}\mid \sqrt{\mid \mathit{bx}\mid }}{\sqrt{\left|\left[\left(\left(2{\lambda }^2+\lambda -1\right)+2\alpha \left(3{\lambda }^2-1\right)\right)b-4p{\lambda }^2{\left(1+\alpha \right)}^2\right]{\mathit{bx}}^2-4\mathit{qa}{\lambda }^2{\left(1+\alpha \right)}^2\right|}}$$E41

$$\mid a_3\mid \le \frac{\mid \mathit{bx}\mid }{\left(3\lambda -1\right)\left(1+2\alpha \right)}+\frac{{\left(\mathit{bx}\right)}^2}{4{\lambda }^2{\left(1+\alpha \right)}^2},$$E42

and for some $\eta \in \mathbb{R}$,

$$\mid a_3-\eta a_2^2\mid \le \left\{\begin{array}{ll}\frac{\mid \mathit{bx}\mid }{\left(3\lambda -1\right)\left(1+2\alpha \right)}& ,\mid \eta -1\mid \le \frac{1}{2\left(3\lambda -1\right)\left(1+2\alpha \right)}\left|A\right|\\ \frac{{\left|\mathit{bx}\right|}^3\mid 1-\eta \mid }{\mid \left[\left(2{\lambda }^2+\lambda -1\right)+2\alpha \left(3{\lambda }^2-1\right)\right]{\left[\mathit{bx}\right]}^2-4{\lambda }^2{\left(1+\alpha \right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }& ,\mid \eta -1\mid \ge \frac{1}{2\left(3\lambda -1\right)\left(1+2\alpha \right)}\left|A\right|\end{array}\right.$$E43

where $A=\left(2{\lambda }^2+\lambda -1\right)+2\alpha \left(3{\lambda }^2-1\right)-\frac{4{\lambda }^2{\left(1+\alpha \right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)}{b^2{x}^2}.$

For $\alpha =0$ in Theorem 3.2, we have the following result.

Corollary 3.3. Let the function $f\in \mathrm{\Sigma }$ given by Eq. (1) be in the class $L\mathrm{\Sigma }\left(\lambda ,x\right)$. Then

$$\mid a_2\mid \le \frac{\mid \mathit{bx}\mid \sqrt{\mid \mathit{bx}\mid }}{\sqrt{\left|\left[\left(2{\lambda }^2+\lambda -1\right)b-4p{\lambda }^2\right]{\mathit{bx}}^2-4\mathit{qa}{\lambda }^2\right|}}$$E44

$$\mid a_3\mid \le \frac{\mid \mathit{bx}\mid }{3\lambda -1}+\frac{{\left(\mathit{bx}\right)}^2}{4{\lambda }^2},$$E45

and for some $\eta \in \mathbb{R}$,

$$\mid a_3-\eta a_2^2\mid \le \left\{\begin{array}{ll}\frac{\mid \mathit{bx}\mid }{3\lambda -1}& ,\mid \eta -1\mid \le \frac{1}{2\left(3\lambda -1\right)}\left|A_0\right|\\ \frac{{\left|\mathit{bx}\right|}^3\mid 1-\eta \mid }{\mid \left(2{\lambda }^2+\lambda -1\right){\left[\mathit{bx}\right]}^2-4{\lambda }^2\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }& ,\mid \eta -1\mid \ge \frac{1}{2\left(3\lambda -1\right)}\left|A_0\right|\end{array}\right.$$E46

where $A_0=\left(2{\lambda }^2+\lambda -1\right)-\frac{4{\lambda }^2\left({\mathit{pbx}}^2+\mathit{qa}\right)}{b^2{x}^2}.$

For $\alpha =1$ Theorem 3.2, we have the following result.

corollary 3.3.

Let the function $f\in \mathrm{\Sigma }$ given by Eq. (1) be in the class $L\mathrm{\Sigma }\left(\lambda ,1,x\right)$. Then

$$\mid a_2\mid \le \frac{\mid \mathit{bx}\mid \sqrt{\mid \mathit{bx}\mid }}{\sqrt{\left|\left[\left(\left(2{\lambda }^2+\lambda -1\right)+2\left(3{\lambda }^2-1\right)\right)b-16p{\lambda }^2\right]{\mathit{bx}}^2-16\mathit{qa}{\lambda }^2\right|}}$$E47

$$\mid a_3\mid \le \frac{\mid \mathit{bx}\mid }{3\left(3\lambda -1\right)}+\frac{{\left(\mathit{bx}\right)}^2}{16{\lambda }^2},$$E48

and for some $\eta \in \mathbb{R}$,

$$\mid a_3-\eta a_2^2\mid \le \left\{\begin{array}{ll}\frac{\mid \mathit{bx}\mid }{3\left(3\lambda -1\right)}& ,\mid \eta -1\mid \le \frac{1}{6\left(3\lambda -1\right)}\left|A_1\right|\\ \frac{{\left|\mathit{bx}\right|}^3\mid 1-\eta \mid }{\mid \left[\left(2{\lambda }^2+\lambda -1\right)+2\left(3{\lambda }^2-1\right)\right]{\left[\mathit{bx}\right]}^2-16{\lambda }^2\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }& ,\mid \eta -1\mid \ge \frac{1}{6\left(3\lambda -1\right)}\left|A_1\right|\end{array}\right.$$E49

where $A_1=\left(2{\lambda }^2+\lambda -1\right)+2\left(3{\lambda }^2-1\right)-\frac{16{\lambda }^2\left({\mathit{pbx}}^2+\mathit{qa}\right)}{b^2{x}^2}.$

For $\lambda =1$ Theorem 3.2, we have the following result.

Corollary 3.4. Let the function $f\in \mathrm{\Sigma }$ given by Eq. (1) be in the class $L\mathrm{\Sigma }\left(1,\alpha ,x\right)$. Then

$$\mid a_2\mid \le \frac{\mid \mathit{bx}\mid \sqrt{\mid \mathit{bx}\mid }}{\sqrt{\left|\left[2\left(1+2\alpha \right)b-4p{\left(1+\alpha \right)}^2\right]{\mathit{bx}}^2-4\mathit{qa}{\left(1+\alpha \right)}^2\right|}}$$E50

$$\mid a_3\mid \le \frac{\mid \mathit{bx}\mid }{2\left(1+2\alpha \right)}+\frac{{\left(\mathit{bx}\right)}^2}{4{\left(1+\alpha \right)}^2},$$E51

and for some $\eta \in \mathbb{R}$,

$$\mid a_3-\eta a_2^2\mid \le \left\{\begin{array}{ll}\frac{\mid \mathit{bx}\mid }{2\left(1+2\alpha \right)}& ,\mid \eta -1\mid \le \frac{1}{4\left(1+2\alpha \right)}\left|B\right|\\ \frac{{\left|\mathit{bx}\right|}^3\mid 1-\eta \mid }{\mid 2\left(1+\alpha \right){\left[\mathit{bx}\right]}^2-4{\left(1+\alpha \right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }& ,\mid \eta -1\mid \ge \frac{1}{4\left(1+2\alpha \right)}\left|B\right|\end{array}\right.$$E52

where $B=2\left(1+2\alpha \right)-\frac{4{\left(1+\alpha \right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)}{b^2{x}^2}.$.

A function $f\in \mathrm{\Sigma }$ given by (1.1) is said to be in the class $M\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$ if the following conditions are satisfied:

$${\left(\frac{2z{\left[f^{\prime }\left(z\right)\right]}^{\lambda }}{f\left(z\right)-f\left(-z\right)}\right)}^{\alpha }{\left(\frac{2{\left[{\left(z{f}^{\prime }\left(z\right)\right)}^{\prime }\right]}^{\lambda }}{{\left[f\left(z\right)-f\left(-z\right)\right]}^{\text{'}}}\right)}^{1-\alpha }\prec \mathrm{\Omega }\left(x,z\right)+1-\alpha$$E53

and

$${\left(\frac{2w{\left[g^{\prime }\left(w\right)\right]}^{\lambda }}{g\left(w\right)-g\left(-w\right)}\right)}^{\alpha }{\left(\frac{2{\left[{\left(w{g}^{\prime }\left(w\right)\right)}^{\prime }\right]}^{\lambda }}{{\left[g\left(w\right)-g\left(-w\right)\right]}^{\prime }}\right)}^{1-\alpha }\prec \mathrm{\Omega }\left(x,w\right)+1-\alpha$$E54

where the real constants $a$, $b$, and $q$ are as in Eq. (25) and $g\left(w\right)=f^{-1}\left(z\right)$ is given by Eq. (4).

Theorem 3.3. Let the function $f\in \mathrm{\Sigma }$ given by Eq. (1) be in the class $M\mathrm{\Sigma }\left(\lambda ,\alpha ,x\right)$. Then

$$\mid a_2\mid \le \frac{\mid \mathit{bx}\mid \sqrt{\mid \mathit{bx}\mid }}{\sqrt{\left|\left[\left(2{\lambda }^2{\left(\alpha -2\right)}^2+\left(\lambda +2\alpha -3\right)\right)b-4p{\lambda }^2{\left(\alpha -2\right)}^2\right]{\mathit{bx}}^2-4\mathit{qa}{\lambda }^2{\left(\alpha -2\right)}^2)\right|}}$$E55

$$\mid a_3\mid \le \frac{\mid \mathit{bx}\mid }{\left(3\lambda -1\right)\left(|3-2\alpha |\right)}+\frac{{\left(\mathit{bx}\right)}^2}{4{\lambda }^2{\left(\alpha -2\right)}^2},$$E56

and for some $\eta \in \mathbb{R}$,

$$\mid a_3-\eta a_2^2\mid \le \left\{\begin{array}{ll}\frac{\mid \mathit{bx}\mid }{\left(3\lambda -1\right)\left(3-2\alpha \right)}& ,\mid \eta -1\mid \le \frac{1}{2\left(3\lambda -1\right)\left(3-2\alpha \right)}\left|C\right|\\ \frac{{\left|\mathit{bx}\right|}^3\mid 1-\eta \mid }{\mid \left[2{\lambda }^2{\left(\alpha -2\right)}^2+\left(\lambda +2\alpha -3\right)\right]{\left[\mathit{bx}\right]}^2-4{\lambda }^2{\left(\alpha -2\right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }& ,\mid \eta -1\mid \ge \frac{1}{2\left(3\lambda -1\right)\left(3-2\alpha \right)}\left|C\right|\end{array}\right.$$E57

where $C=2{\lambda }^2{\left(\alpha -2\right)}^2+\left(\lambda +2\alpha -3\right)-\frac{4{\lambda }^2{\left(\alpha -2\right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)}{b^2{x}^2}.$

For $\lambda =1$ Theorem 3.3, we have the following result.

Corollary 3.5. Let the function $f\in \mathrm{\Sigma }$ given by Eq. (1) be in the class $M\mathrm{\Sigma }\left(1,\alpha ,x\right)$. Then

$$\mid a_2\mid \le \frac{\mid \mathit{bx}\mid \sqrt{\mid \mathit{bx}\mid }}{\sqrt{\left|\left[\left(2{\left(\alpha -2\right)}^2+2\left(\alpha -1\right)\right)b-4p{\left(\alpha -2\right)}^2\right]{\mathit{bx}}^2-4\mathit{qa}{\left(\alpha -2\right)}^2)\right|}}$$E58

$$\mid a_3\mid \le \frac{\mid \mathit{bx}\mid }{2\left(|3-2\alpha |\right)}+\frac{{\left(\mathit{bx}\right)}^2}{4{\left(\alpha -2\right)}^2},$$E59

and for some $\eta \in \mathbb{R}$,

$$\mid a_3-\eta a_2^2\mid \le \left\{\begin{array}{ll}\frac{\mid \mathit{bx}\mid }{2\left(3-2\alpha \right)}& ,\mid \eta -1\mid \le \frac{1}{4\left(3-2\alpha \right)}\left|C_1\right|\\ \frac{{\left|\mathit{bx}\right|}^3\mid 1-\eta \mid }{\mid \left[2{\left(\alpha -2\right)}^2+2\left(\alpha -1\right)\right]{\left[\mathit{bx}\right]}^2-4{\left(\alpha -2\right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)\mid }& ,\mid \eta -1\mid \ge \frac{1}{4\left(3-2\alpha \right)}\left|C_1\right|\end{array}\right.$$E60

where $C_1=2{\left(\alpha -2\right)}^2+2\left(\alpha -1\right)-\frac{4{\left(\alpha -2\right)}^2\left({\mathit{pbx}}^2+\mathit{qa}\right)}{b^2{x}^2}.$

2. Conclusions

The study conducted in this chapter involves certain subclasses of bi-univalent functions examined by using Gegenbauer polynomials and Hordam polynomials, respectively, in the open unit disc. The main results are contained in which coefficient estimates are obtained for each subclass of bi-univalent functions, which could inspire researchers to focus on other aspects such as certain families of bi-univalent functions using other orthogonal polynomials.

Acknowledgments

It is a great pleasure for me to appreciate Prof. Faruk Özge and academic editor for their precious time in reviewing this chapter and putting it in their book project.