Coefficient Estimates for New Subclasses of Bi-Univalent Functions with Bounded Boundary Rotation by Using Faber Polynomial Technique

Abstract

In this article, the authors use the Faber polynomial expansions to find the general coefficient estimates for a few new subclasses of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Some of the results improve the existing coefficient bounds in the literature.

1. Introduction and Definitions

Let $\mathcal{A}$ denote the class of functions f given by

$$f\left(z\right)=z+\sum _{m=2}^{\infty }{b}_m{z}^m,$$

(1)

which are analytic on the open unit disk $\mathbb{D}=\{z\in \mathbb{C}:|z|<1\}$ and normalized by $f\left(0\right)=0$ and $f^{\prime }\left(0\right)=1.$ Let us consider $\mathcal{S}\subset \mathcal{A}$ consisting of univalent functions in $\mathbb{D}.$ A function $f\in \mathcal{S}$ is said to be starlike if f maps the unit disk onto a starlike region $G,$ i.e.,

$$w\in f\left(G\right)\iff tw\in f\left(\mathbb{D}\right),\forall 0\le t\le 1.$$

A function $f\in \mathcal{C}$ is said to be convex if f maps the unit disk onto a convex

$$w_1,w_2\in f\left(G\right)\iff t_1{w}_1+(1-t_2)w_2\in f\left(\mathbb{D}\right),\forall 0\le t_1,t_2\le 1.$$

The analytic criterions for the class of starlike functions of order $\eta$ and the class of convex functions of order $\eta$ are defined by

$${\mathcal{S}}^{*}\left(\eta \right)=\left\{f\in \mathcal{S}:\Re \left({\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\right)>\eta ,0\le \eta <1andz\in \mathbb{D}\right\}$$

and

$$\mathcal{C}\left(\eta \right)=\left\{f\in \mathcal{S}:\Re \left(1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\right)>\eta ,0\le \eta <1andz\in \mathbb{D}\right\},$$

respectively.

The Koebe one-quarter theorem ensures that a function $f\in \mathcal{S}$ such that

$$f\left(\mathbb{D}\right)\supseteq \left\{w:\left|w\right|<{\displaystyle \frac{1}{4}}\right\}.$$

Hence, every function $f\in \mathcal{S}$ given in the form Equation (1) has an inverse $g=f^{-1},$ such that

$$g\left(f\left(z\right)\right)=z,z\in \mathbb{D}andg\left(f\left(w\right)\right)=w,\left|w\right|<\rho ,\rho \ge {\displaystyle \frac{1}{4}}.$$

Thus, the inverse function $g=f^{-1}$ given by

$$g\left(w\right)=f^{-1}\left(w\right)=w-b_2{w}^2+(2b_2^2-b_3)w^3-(5b_2^3-5b_2{b}_3+b_4)w^4+\cdots .$$

(2)

If both f and its inverse $g=f^{-1}$ are univalent in $\mathbb{D},$ then f is called bi-univalent. Let $\mathsf{\Sigma }$ denote the class of all bi-univalent functions. The concept of class $\mathsf{\Sigma }$ originates from Lewin [1] in 1967 where he proved that $\left|b_2\right|\le 1.51.$ Many well-known mathematicians Styer and Wright [2], Brannan and Taha [3] and Srivastava et al. [4] investigated the class $\mathsf{\Sigma }.$ Brannan and Taha [3] introduced the classes of bi-starlike functions of order $\eta$ and bi-convex functions of order $\eta$ denoted by ${\mathcal{S}}_{\mathsf{\Sigma }}^{*}\left(\eta \right)$ and ${\mathcal{C}}_{\mathsf{\Sigma }}\left(\eta \right)$, respectively. Styer and Wright [2] prove that $|b_2|>{\displaystyle \frac{4}{3}}.$ The class $\mathsf{\Sigma }$ is a non-empty class as the functions ${\displaystyle f_1\left(z\right)=z+\frac{z^3}{6}}$ and ${\displaystyle f_2\left(z\right)=z-\frac{z^3}{9}}$ belong to the class $\mathsf{\Sigma }$ apart from the functions pointed out by Srivastava et al. [4]. In recent times, Breaz et al. [5] and Sharma et al. [6] introduced certain subclasses of bi-univalent functions with bounded boundary rotation. Breaz et al. [5] introduced the class of bi-close-to-convex functions with bounded boundary rotation of order $\eta$ and Sharma et al. [6] introduced the class of bi-quasi-convex and exponentially bi-convex functions with bounded boundary rotation of order $\eta$ which were denoted by ${\mathcal{K}}_{\mathsf{\Sigma }}(\mu ,\eta ),$ ${\mathcal{Q}}_{\mathsf{\Sigma }}^{*}(\mu ,\eta )$ and ${\mathcal{E}}_{\mathsf{\Sigma }}(\mu ,\eta )$, respectively. In the works mentioned above, the authors have obtained initial coefficient bounds for the new classes that they introduced in their articles.

Let ${\mathcal{V}}_{\mu }$ denote the class of functions given in the form (1) analytic in the open unit disc $\mathbb{D}$ which satisfy the condition $f^{\prime }\left(z\right)\ne 0$ for all z in $\mathbb{D},$ mapping $\mathbb{D}$ onto a domain with boundary rotation bounded by $\mu \pi .$ Paatero [7] has shown that $f\in {\mathcal{V}}_{\mu }$ such that

$$f^{\prime }\left(z\right)=exp\left\{-{\int }_0^{2\pi }log|1-z{e}^{-i\zeta }|d\psi \left(\zeta \right)\right\},$$

where $\psi \left(\zeta \right)$ is a real-valued function of bounded variation with

$${\int }_0^{2\pi }d\psi \left(\zeta \right)=2and{\int }_0^{2\pi }d\psi \left(\zeta \right)\le \mu .$$

(3)

Clearly, $f\in {\mathcal{V}}_{\mu }$ if and only if $f\left(z\right)$ is a normalized univalent function mapping $\mathbb{D}$ onto a convex domain. Furthermore, Paatero [7] has shown that for $\mu \in [2,4]$ the classes ${\mathcal{V}}_{\mu }$ consist entirely of univalent functions. Let ${\mathcal{V}}_{\mu }\left(\eta \right)$ denote the class of all analytic functions of a bounded boundary rotation of order $\eta (0\le \eta <1).$

Let ${\mathcal{R}}_{\mu }$ denote the class of functions given in the form Equation (1) analytic in the open unit disc $\mathbb{D}$ which satisfy the condition $f^{\prime }\left(z\right)\ne 0$ for all z in $\mathbb{D},$ and which map $\mathbb{D}$ onto a domain with radius rotation bounded by $\mu \pi .$ Pinchuk [8] has shown that $f\in {\mathcal{R}}_{\mu }$ such that

$$f\left(z\right)=exp\left\{-{\int }_0^{2\pi }log|1-z{e}^{-i\zeta }|d\psi \left(\zeta \right)\right\},$$

where $\psi \left(\zeta \right)$ is introduced in Equation (3). Clearly, $f\in {\mathcal{R}}_{\mu }$ if and only if $f\left(z\right)$ is a normalized univalent function mapping $\mathbb{D}$ onto a starlike domain. Let ${\mathcal{R}}_{\mu }\left(\eta \right)$ denote the class of all analytic functions of a bounded radius rotation of order $\eta (0\le \eta <1).$

Let ${\mathcal{P}}_{\mu }$ denote the class of functions given in the form

$$ħ\left(z\right)=1+\sum _{m=1}^{\infty }{ħ}_m{z}^m=1+{ħ}_{1}z+{ħ}_2{z}^2+\cdots ,$$

(4)

analytic in the open unit disc $\mathbb{D}$ and satisfying the conditions

$${\int }_0^{2\pi }\Re \left|ħ\left(r{e}^{i\zeta }\right)\right|d\zeta \le \mu \pi .$$

The class ${\mathcal{P}}_{\mu }$ generalizes the class of analytic functions with a positive real part in open unit disk $\mathbb{D}.$ Indeed, Pinchuk [8] has shown that $ħ\in {\mathcal{P}}_{\mu }$ such that

$$ħ\left(z\right)={\int }_0^{2\pi }{\displaystyle \frac{1+z{e}^{-i\zeta }}{1-z{e}^{-i\zeta }}}d\psi \left(\zeta \right),$$

where $\psi \left(\zeta \right)$ is introduced in Equation (3). Let ${\mathcal{P}}_{\mu }\left(\eta \right)$ denote the class of functions given in the form Equation (4) and satisfying the condition:

$${\int }_0^{2\pi }\Re \left|{\displaystyle \frac{ħ\left(z\right)-\eta }{1-\eta }}\right|d\zeta \le \mu \pi .$$

The classes ${\mathcal{V}}_{\mu }\left(\eta \right),$ ${\mathcal{R}}_{\mu }\left(\eta \right)$ and ${\mathcal{P}}_{\mu }\left(\eta \right)$ were introduced by Padmanabhan and Parvatham [9]. Pinchuk [8] gave interesting relation connections between ${\mathcal{V}}_{\mu },$ ${\mathcal{R}}_{\mu }$ and ${\mathcal{P}}_{\mu }$ and are given by

$$f\in {\mathcal{V}}_{\mu }\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}z{f}^{\prime }\in {\mathcal{R}}_{\mu },$$

$$f\in {\mathcal{V}}_{\mu }\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}1+{\displaystyle \frac{z{f}^{″}}{f^{\prime }}}\in {\mathcal{P}}_{\mu }$$

and

$$f\in {\mathcal{R}}_{\mu }\mathrm{if}\mathrm{and}\mathrm{only}\mathrm{if}{\displaystyle \frac{z{f}^{\prime }}{f}}\in {\mathcal{P}}_{\mu }.$$

Lemma 1

([10]). If the analytic function ℏ given in (4) is such that $ħ\in {\mathcal{P}}_{\mu }\left(\eta \right)$, then

$$|{ħ}_m|\le \mu (1-\eta ),m\in \mathbb{N}.$$

Definition 1.

Let $0\le \eta <1$ and $2\le \mu \le 4.$ A function $f\in \mathsf{\Sigma }$ is said to be in the class ${\mathcal{R}}_{\mathsf{\Sigma }}(k,\eta )$ if f and its inverse $g=f^{-1}$ satisfies the conditions:

$${\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}\in {\mathcal{P}}_{\mu }\left(\eta \right)$$

and

$${\displaystyle \frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}}\in {\mathcal{P}}_{\mu }\left(\eta \right).$$

The class ${\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )$ was investigated by Li et al. [11].

Definition 2.

Let $0\le \eta <1$ and $2\le \mu \le 4.$ A function $f\in \mathsf{\Sigma }$ is said to be in the class ${\mathcal{V}}_{\mathsf{\Sigma }}(k,\eta )$ if f and its inverse $g=f^{-1}$ satisfies the conditions:

$$1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}\in {\mathcal{P}}_{\mu }\left(\eta \right)$$

and

$$1+{\displaystyle \frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}}\in {\mathcal{P}}_{\mu }\left(\eta \right).$$

The class ${\mathcal{V}}_{\mathsf{\Sigma }}(\mu ,\eta )$ was investigated by Li et al. [11].

Definition 3.

Let $0\le \delta \le 1,$ $2\le \mu \le 4$ and $0\le \eta <1.$ A function $f\in \mathsf{\Sigma }$ is said to be in the class ${\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ]$ if f and its inverse $g=f^{-1}$ satisfies the conditions:

$$(1-\delta ){\displaystyle \frac{f\left(z\right)}{z}}+\delta f^{\prime }\left(z\right)\in {\mathcal{P}}_{\mu }\left(\eta \right)$$

and

$$(1-\delta ){\displaystyle \frac{g\left(w\right)}{w}}+\delta g^{\prime }\left(w\right)\in {\mathcal{P}}_{\mu }\left(\eta \right).$$

The class ${\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ]$ was investigated by Sharma et al. [12].

Definition 4.

Let $\lambda \ge 0,$ $2\le \mu \le 4$ and $0\le \eta <1.$ A function $f\in \mathsf{\Sigma }$ is said to be in the class ${\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )$ if f and its inverse $g=f^{-1}$ satisfies the conditions:

$$f^{\prime }\left(z\right)+\lambda f^{″}\left(z\right)\in {\mathcal{P}}_{\mu }\left(\eta \right)$$

and

$$g^{\prime }\left(w\right)+\lambda f^{″}\left(z\right)\in {\mathcal{P}}_{\mu }\left(\eta \right).$$

The class ${\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )$ was investigated by Sharma et al. [12].

Remark 1.

(i) For the choice of $\mu =2$ in Definition 1, ${\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )\equiv {\mathcal{S}}_{\mathsf{\Sigma }}^{*}\left(\eta \right)$ introduced in [3].

(ii) For the choice of $\mu =2$ in Definition 2, ${\mathcal{V}}_{\mathsf{\Sigma }}(\mu ,\eta )\equiv {\mathcal{C}}_{\mathsf{\Sigma }}\left(\eta \right)$ introduced in [3].

(iii) For $\delta =1$ in Definition 3 (or for $\lambda =0$ in Definition 4), the class ${\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ]\equiv {\mathcal{N}}_{\mathsf{\Sigma }}(\mu ,\eta )$ (or the class ${\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )\equiv {\mathcal{N}}_{\mathsf{\Sigma }}(\mu ,\eta )$) introduced in [12].

(iv) For the choice of $\mu =2$ in Definition 3, ${\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ]\equiv {\mathcal{B}}_{\mathsf{\Sigma }}(\eta ,\lambda )$ introduced in [13].

(v) For the choice of $\mu =2$ in Definition 4, ${\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )\equiv {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }\left(\eta \right)$, the class involving complex order introduced in [14].

2. Preliminaries Results

Faber polynomials allow several important generalizations of results in every branch of research as well as an important role in various fields of mathematical sciences, especially in geometric function theory. Using the Faber polynomial expansion, for functions $f\in \mathsf{\Sigma }$ of the form Equation (1), Airault and Bouali [15] expressed the coefficients of $g\left(w\right)$ as

$$g\left(w\right)=f^{-1}\left(w\right)=w+\sum _{m=2}^{\infty }{\displaystyle \frac{1}{m}}{K}_{m-1}^{-m}(b_2,b_3,\cdots ,b_m)w^m,$$

where

$$\begin{array}{cc}\hfill K_{m-1}^{-m}(b_2,b_3,\cdots ,b_m)=& {\displaystyle \frac{(-m)!}{(-2m+1)!(m-1)!}}{b}_2^{m-1}\hfill \\ \hfill +& {\displaystyle \frac{(-m)!}{\left(2\right(-m+1\left)\right)!(m-1)!}}{b}_2^{n-3}{b}_3\hfill \\ \hfill +& {\displaystyle \frac{(-m)!}{(-2m+3)!(m-4)!}}{b}_2^{m-4}{b}_4\hfill \\ \hfill +& {\displaystyle \frac{(-m)!}{\left(2(-m+2)\right)!(m-5)!b_2^{m-5}}}[b_5+(-m+2)b_3^2]\hfill \\ \hfill +& \sum _{j\ge 7}{b}_2^{m-j}{V}_j\hfill \end{array}$$

such that $V_j$ with $7\le j\le m$ is a homogeneous polynomial in the variables $b_2,b_3,\cdots ,b_m.$ In particular, the first three terms of $K_{m-1}^{-m}$ are

$${\displaystyle \frac{1}{2}}{K}_1^{-2}=-b_2$$

$${\displaystyle \frac{1}{3}}{K}_2^{-3}=2b_2^2-b_3$$

and

$${\displaystyle \frac{1}{4}}{K}_3^{-4}=-(5b_2^3-5b_2{b}_3+b_4).$$

In general, an expansion of $K_m^q$ is given by

$$K_m^q=q{b}_m+{\displaystyle \frac{q(q-1)}{2}}{B}_m^2+{\displaystyle \frac{q!}{(q-3)!3!}}{B}_m^3+\cdots +{\displaystyle \frac{q!}{(q-m)!m!}}{B}_m^m,$$

where $B_m^q=B_m^q(b_2,b_3,\cdots ,b_m)$ and defined by

$$B_m^m(b_2,b_3,\dots ,b_m)=\sum _{k=1}^{\infty }{\displaystyle \frac{k!{\left(b_2\right)}^{{\tau }_1}\dots {\left(b_m\right)}^{{\tau }_{m-1}}}{{\tau }_1\dots {\tau }_{m-1}}},$$

where the sum is taken over all non-negative integers ${\tau }_1,\dots ,{\tau }_m$ satisfying the conditions:

$$\left\{\begin{array}{c}{\tau }_1+{\tau }_2+\cdots +{\tau }_{m-1}=k,\hfill \\ {\tau }_1+2{\tau }_2+\cdots +(m-1){\tau }_{m-1}=m-1.\hfill \end{array}\right.$$

It is clear that

$$B_{m-1}^{m-1}(b_2,b_3,\cdots ,b_m)=b_2^{m-1}.$$

Airault and Bouali [15], showed that for a function $f\in \mathcal{A}$

$${\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}=1-\sum _{m=0}^{\infty }{H}_{m-1}(b_2,b_3,\dots ,b_m)z^{m-1},$$

(5)

where

$$H_{m-1}(b_2,b_3,\dots ,b_m)=\sum _{j_1+2j_2+\cdots +(m-1)j_{m-1}=m-1}^{\infty }\mathcal{B}(j_1,j_2,\dots ,j_{m-1})\left(b_2^{j_1}{b}_3^{j_2}\dots b_m^{j_{m-1}}\right)$$

and

$$\begin{gathered} \mathcal{B}(j_1,j_2,\dots ,j_{m-1})=\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ {(-1)}^{(m-1)+2j_1+\cdots +m{j}_{m-1}}\times {\displaystyle \frac{(j_1+j_2+\cdots +j_{m-1}-1)!(m-1)}{\left(j_1\right)!\left(j_2\right)!\dots (j_{m-1}!)}}. \end{gathered}$$

The first four terms of the Faber polynomial ${\mathcal{H}}_m,$ $m\ge 2$ are given by

$${\mathcal{H}}_1=b_2,$$

$${\mathcal{H}}_2=b_2^2-2b_3,$$

$${\mathcal{H}}_3=-b_2^3+3b_2{b}_3-3b_4$$

and

$${\mathcal{H}}_4=b_2^4-4b_2^2{b}_3+4b_2{b}_4+2b_3^2-4b_5.$$

The Faber polynomials was introduced by Faber [16] in 1903. Gong [17] discussed the importance of the Faber polynomial (for more details, see [18]). Hamidi and Jahangiri [19,20,21] used the Faber polynomial expansion method to find coefficient bounds $|b_m|$ for $m\ge 3$, for many subclasses of bi-univalent functions.

In this article, we use the Faber polynomial expansions to find new general coefficient estimates for well-known classes of bi-univalent functions with bounded boundary rotation and bounded radius rotation. Apart from the interesting new results mentioned, the results also improve the existing coefficient bounds that are available in the literature.

3. Main Results

Theorem 1.

Let $2\le \mu \le 4$, $0\le \eta <1$, $f\in {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )$ and $g\in {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta ).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{\mu (1-\eta )}{m-1}},\forall m\ge 3.$$

(6)

Proof. 

Since $f,g\in {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )$, according to Definition 1, there exists two functions $u\left(z\right)$ and $v\left(w\right)$ belonging to ${\mathcal{P}}_{\mu }\left(\eta \right)$ such that

$${\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}=u\left(z\right)$$

(7)

and

$${\displaystyle \frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}}=v\left(w\right),$$

(8)

where

$$u\left(z\right)=1+\sum _{m=1}^{\infty }{u}_m{z}^m$$

(9)

and

$$v\left(z\right)=1+\sum _{m=1}^{\infty }{v}_m{z}^m.$$

(10)

Therefore, using Faber polynomial we obtain

$${\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(u_1,u_2,\dots ,u_m)z^m$$

(11)

and

$${\displaystyle \frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}}=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(v_1,v_2,\dots ,v_m)w^m.$$

(12)

For any function f given by Equation (1), we obtain

$${\displaystyle \frac{z{f}^{\prime }\left(z\right)}{f\left(z\right)}}=1-\sum _{m=2}^{\infty }{\mathcal{H}}_{m-1}^m(b_2,b_3,\dots ,b_m)z^{m-1}$$

(13)

and

$${\displaystyle \frac{w{g}^{\prime }\left(w\right)}{g\left(w\right)}}=1-\sum _{m=2}^{\infty }{\mathcal{H}}_{m-1}^m(d_2,d_3,\dots ,d_m)w^{m-1},$$

(14)

where ${\displaystyle d_m=\frac{1}{m}{\mathcal{K}}_{m-1}^{-m}(b_2,b_3,\dots ,b_m).}$ Hence, from Equations (11) and (13), we obtain

$${\mathcal{H}}_{m-1}^m(b_2,b_3,\dots ,b_m)={\mathcal{K}}_m^1(u_1,u_2,\dots ,u_m).$$

(15)

Similarly, from Equations (12) and (14), we obtain

$${\mathcal{H}}_{m-1}^m(d_2,d_3,\dots ,d_m)={\mathcal{K}}_m^1(v_1,v_2,\dots ,v_m).$$

(16)

Now, for $b_n=0;$ $2\le n\le m-1,$ we have $d_m=-b_m$ and

$$(m-1)b_m=u_{m-1}$$

(17)

and

$$-(m-1)b_m=v_{m-1}.$$

(18)

Hence, we have

$$b_m={\displaystyle \frac{u_{m-1}}{m-1}}=-{\displaystyle \frac{v_{m-1}}{m-1}}.$$

(19)

Using Lemma 1 in Equation (19), we obtain Equation (6). This completes the proof of Theorem 1. □

For $\mu =2$ in Theorem 1, Corollary 1 verifies the following bound obtained by Hamidi and Jahangiri [20].

Corollary 1.

For $0\le \eta <1$, let $f\in {\mathcal{R}}_{\mathsf{\Sigma }}(2,\eta )\equiv {\mathcal{S}}_{\mathsf{\Sigma }}^{*}\left(\eta \right)$ and $g\in {\mathcal{R}}_{\mathsf{\Sigma }}(2,\eta )\equiv {\mathcal{S}}_{\mathsf{\Sigma }}^{*}\left(\eta \right).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{2(1-\eta )}{m-1}},\forall m\ge 3.$$

Theorem 2.

For $2\le \mu \le 4$ and $0\le \eta <1$, let $f\in {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )$ and $g\in {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta ).$ Then

$$|b_2|\le \sqrt{\mu (1-\eta )},$$

(20)

$$|b_3|\le \mu (1-\eta )$$

(21)

and

$$|b_3-\Delta b_2^2|\le \left\{\begin{array}{cc}\mu (1-\eta )|1-\Delta |:\hfill & {\displaystyle |1-\Delta |\ge \frac{1}{2},}\hfill \\ {\displaystyle \frac{\mu (1-\eta )}{2}:}\hfill & {\displaystyle 0\le |1-\Delta |\le \frac{1}{2}.}\hfill \end{array}\right.$$

(22)

Proof. 

For $m=3$ in Equations (15) and (16), respectively, we obtain

$$2b_3-a_2^2=u_2$$

(23)

and

$$3b_2^2-2b_3=v_2.$$

(24)

Adding Equations (23) and (24), we obtain

$$2b_2^2=u_2+v_2.$$

(25)

Using Lemma 1 in Equation (25), we obtain Equation (20). Subtracting Equation (24) from Equation (23) and by an application of Equation (25), we get

$$4b_3=3u_2+v_2.$$

(26)

Using Lemma 1 in Equation (26), we obtain Equation (21). For some real number $\Delta$ and from Equations (25) and (26), we obtain

$$b_3-\Delta b_2^2={\displaystyle \frac{1}{2}}\left[\left(\mathsf{\Psi }(\Delta )+{\displaystyle \frac{1}{2}}\right)u_2+\left(\mathsf{\Psi }(\Delta )-{\displaystyle \frac{1}{2}}\right)v_2\right],$$

(27)

where $\mathsf{\Psi }(\Delta )=1-\Delta .$ Using Lemma 1 in Equation (27), we obtain Equation (22). This completes the proof of Theorem 2. □

Remark 2.

(i) Theorem 2 verifies the bounds of $|b_2|$ and $|b_3|$ obtained by Sharma et al. [12].

(ii) The choice of $\mu =2$ in Theorem 2 verifies the bounds of $|b_2|$ and $|b_3|$ obtained by Brannan and Taha [3].

Theorem 3.

For $\mu \ge 2$ and $0\le \eta <1$, let $f\in {\mathcal{V}}_{\mathsf{\Sigma }}(\mu ,\eta )$ and $g\in {\mathcal{V}}_{\mathsf{\Sigma }}(\mu ,\eta ).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{\mu (1-\eta )}{m(m-1)}},\forall m\ge 3.$$

(28)

Proof. 

Since $f,g\in {\mathcal{V}}_{\mathsf{\Sigma }}(\mu ,\eta )$, by Definition 2, there exists two functions $u\left(z\right)$ and $v\left(w\right)$ belonging to ${\mathcal{P}}_{\mu }\left(\eta \right)$ such that

$$1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}=u\left(z\right)$$

(29)

and

$$1+{\displaystyle \frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}}=v\left(w\right),$$

(30)

where $u\left(z\right)$ and $v\left(w\right)$ are given in the form Equations (9) and (10). Therefore, using the Faber polynomial, we obtain

$$1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(u_1,u_2,\dots ,u_m)z^m$$

(31)

and

$$1+{\displaystyle \frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}}=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(v_1,v_2,\dots ,v_m)w^m.$$

(32)

For any function f given by Equation (1), we obtain

$$1+{\displaystyle \frac{z{f}^{″}\left(z\right)}{f^{\prime }\left(z\right)}}=1-\sum _{m=2}^{\infty }{\mathcal{H}}_{m-1}^m(2b_2,3b_3,\dots ,m{b}_m)z^{m-1}$$

(33)

and

$$1+{\displaystyle \frac{w{g}^{″}\left(w\right)}{g^{\prime }\left(w\right)}}=1-\sum _{m=2}^{\infty }{\mathcal{H}}_{m-1}^m(2d_2,3d_3,\dots ,m{d}_m)w^{m-1},$$

(34)

where ${\displaystyle d_m=\frac{1}{m}{\mathcal{K}}_{m-1}^{-m}(2b_2,3b_3,\dots ,m{b}_m).}$ Hence, from Equations (31) and (33), we obtain

$${\mathcal{H}}_{m-1}^m(2b_2,3b_3,\dots ,m{b}_m)={\mathcal{K}}_m^1(u_1,u_2,\dots ,u_m).$$

(35)

Similarly, from Equations (32) and (34), we obtain

$${\mathcal{H}}_{m-1}^m(2d_2,3d_3,\dots ,m{d}_m)={\mathcal{K}}_m^1(v_1,v_2,\dots ,v_m).$$

(36)

Now, for $b_n=0;$ $2\le n\le m-1,$ we have $d_m=-b_m$ and

$$m(m-1)b_m=u_{m-1}$$

(37)

and

$$-m(m-1)b_m=v_{m-1}.$$

(38)

Hence, we have

$$b_m={\displaystyle \frac{u_{m-1}}{m(m-1)}}=-{\displaystyle \frac{v_{m-1}}{m(m-1)}}.$$

(39)

Using Lemma 1 in Equation (39), we obtain Equation (28). This completes the proof of Theorem 3. □

For $\mu =2$ in Theorem 3, Corollary 2 verifies the bound obtained by Hamidi and Jahangiri [20] and is stated as follows.

Corollary 2.

For $0\le \eta <1$, let $f\in {\mathcal{V}}_{\mathsf{\Sigma }}(2,\eta )\equiv {\mathcal{C}}_{\mathsf{\Sigma }}\left(\eta \right)$ and $g\in {\mathcal{V}}_{\mathsf{\Sigma }}(2,\eta )\equiv {\mathcal{C}}_{\mathsf{\Sigma }}\left(\eta \right).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{2(1-\eta )}{m(m-1)}},\forall m\ge 3.$$

Theorem 4.

Let $\mu \ge 2$, $0\le \eta <1$, $f\in {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )$ and $g\in {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )$. Then

$$|b_2|\le \sqrt{{\displaystyle \frac{\mu (1-\eta )}{2}}},$$

(40)

$$|b_3|\le {\displaystyle \frac{\mu (1-\eta )}{2}}$$

(41)

and

$$|b_3-\Delta b_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\mu (1-\eta )}{2}|1-\Delta |:}\hfill & {\displaystyle |1-\Delta |\ge \frac{1}{3},}\hfill \\ {\displaystyle \frac{\mu (1-\eta )}{6}:}\hfill & {\displaystyle 0\le |1-\Delta |\le \frac{1}{3}.}\hfill \end{array}\right.$$

(42)

Proof. 

For $m=3$ in Equations (35) and (36), respectively, we obtain,

$$6b_3-4a_2^2=u_2$$

(43)

and

$$8b_2^2-6b_3=v_2.$$

(44)

Adding Equations (43) and (44), we obtain

$$4b_2^2=u_2+v_2.$$

(45)

Using Lemma 1 in Equation (45), we obtain Equation (40). Subtracting Equation (44) from Equation (43) and by an application of (45), we obtain

$$6b_3=2u_2+v_2.$$

(46)

Using Lemma 1 in Equation (46), we obtain Equation (41). For some real number $\Delta$ and from Equations (45) and (46), we obtain

$$b_3-\Delta b_2^2={\displaystyle \frac{1}{4}}\left[\left(\mathsf{\Psi }(\Delta )+{\displaystyle \frac{1}{3}}\right)u_2+\left(\mathsf{\Psi }(\Delta )-{\displaystyle \frac{1}{3}}\right)v_2\right],$$

(47)

where $\mathsf{\Psi }(\Delta )=1-\Delta .$ Using Lemma 1 in Equation (47), we obtain Equation (42). This completes the proof of Theorem 4. □

Remark 3.

(i) Theorem 4 verifies the bounds of $|b_2|$ and $|b_3|$ obtained by Sharma et al. [12].

(ii) For $\mu =2$, Theorem 4 verifies the bounds of $|b_2|$ and $|b_3|$ obtained by Brannan and Taha [3].

Theorem 5.

For $2\le \mu \le 4,$ $0\le \eta <1$ and $\delta \ge 1,$ let $f\in {\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ]$ and $g\in {\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ].$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{\mu (1-\eta )}{1+\delta (m-1)}},\forall m\ge 3.$$

(48)

Proof. 

Since $f,g\in {\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ]$, by Definition 3, there exist functions $u\left(z\right)$ and $v\left(w\right)$ belonging to ${\mathcal{P}}_{\mu }\left(\eta \right)$ such that

$$(1-\delta ){\displaystyle \frac{f\left(z\right)}{z}}+\delta f^{\prime }\left(z\right)=u\left(z\right)$$

(49)

and

$$(1-\delta ){\displaystyle \frac{g\left(w\right)}{w}}+\delta g^{\prime }\left(w\right)=v\left(w\right),$$

(50)

where $u\left(z\right)$ and $v\left(w\right)$ are given in the form Equations (9) and (10). Therefore, using the Faber polynomial, we obtain

$$(1-\delta ){\displaystyle \frac{f\left(z\right)}{z}}+\delta f^{\prime }\left(z\right)=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(u_1,u_2,\dots ,u_m)z^m$$

(51)

and

$$(1-\delta ){\displaystyle \frac{g\left(w\right)}{w}}+\delta g^{\prime }\left(w\right)=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(v_1,v_2,\dots ,v_m)w^m.$$

(52)

For any function f given by Equation (1), we obtain

$$(1-\delta ){\displaystyle \frac{f\left(z\right)}{z}}+\delta f^{\prime }\left(z\right)=1+\sum _{m=2}^{\infty }[1+\delta (m-1)]b_m{z}^{m-1}$$

(53)

and for its analytic continuation map $g=f^{-1},$ we have

$$\begin{gathered} (1-\delta )\frac{g\left(w\right)}{w}+\delta g^{\prime }\left(w\right)=1+\sum _{m=1}^{\infty }[1+\delta (m-1)]d_m{w}^{m-1}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \\ =1+\sum _{m=1}^{\infty }[1+\delta (m-1)]{\displaystyle \frac{1}{m}}{\mathcal{K}}_{m-1}^{-m}(b_2,b_3,\dots ,b_m)w^{m-1}. \end{gathered}$$

(54)

Hence, from Equations (51) and (53), we obtain

$$[1+\delta (m-1)]b_m={\mathcal{K}}_{m-1}^1(u_1,u_2,\dots ,u_{m-1}).$$

(55)

Similarly, from Equations (52) and (54), we obtain

$${\displaystyle \frac{1}{m}}[1+\delta (m-1)]{\mathcal{K}}_{m-1}^{-m}(b_0,b_1,\dots ,b_m)={\mathcal{K}}_{m-1}^1(v_1,v_2,\dots ,v_{m-1}).$$

(56)

Now for $b_n=0;$ $2\le n\le m-1,$ we have $d_m=-b_m$ and

$$[1+\delta (m-1)]b_m=u_{m-1}$$

(57)

and

$$-[1+\delta (m-1)]b_m=v_{m-1}.$$

(58)

Hence, we have

$$b_m={\displaystyle \frac{u_{m-1}}{1+\delta (m-1)}}=-{\displaystyle \frac{v_{m-1}}{1+\delta (m-1)}}.$$

(59)

Using Lemma 1 in Equation (59), we obtain Equation (48). This completes the proof of Theorem 5. □

For $\mu =2$ in Theorem 5, we obtain the following Corollary 3, which essentially verifies the bound obtained by Jahangiri and Hamidi [22].

Corollary 3.

Let $0\le \eta <1$ and $\delta \ge 1.$ Furthermore, let $f\in {\mathcal{Q}}_{\mathsf{\Sigma }}^{\delta }(2,\eta )\equiv \mathcal{D}(\eta ,\delta )$ and $g\in {\mathcal{Q}}_{\mathsf{\Sigma }}^{\delta }(2,\eta )\equiv \mathcal{D}(\eta ,\delta ).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{2(1-\eta )}{1+\delta (m-1)}},\forall m\ge 3.$$

For $\delta =1$ in Theorem 5, we obtain the following corollary.

Corollary 4.

For $2\le \mu \le 4$ and $0\le \eta <1$, let $f\in {\mathcal{K}}_{\mathsf{\Sigma }}^1[\mu ,\eta ]\equiv {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta )$ and $g\in {\mathcal{K}}_{\mathsf{\Sigma }}^1[\mu ,\eta ]\equiv {\mathcal{R}}_{\mathsf{\Sigma }}(\mu ,\eta ).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{\mu (1-\eta )}{m}},\forall m\ge 3.$$

Remark 4.

For $\mu =2$ in Corollary 4, we obtain the bound obtained by Hamidi and Jahangiri [20].

Theorem 6.

For $2\le \mu \le 4$ and $0\le \eta <1$, let $f\in {\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ]$ and $g\in {\mathcal{K}}_{\mathsf{\Sigma }}^{\delta }[\mu ,\eta ].$ Then

$$|b_2|\le min\left\{{\displaystyle \frac{\mu (1-\eta )}{1+\delta }},\sqrt{{\displaystyle \frac{\mu (1-\eta )}{1+2\delta }}}\right\},$$

(60)

$$|b_3|\le {\displaystyle \frac{\mu (1-\eta )}{1+2\delta }}$$

(61)

and

$$|b_3-\Delta b_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\mu (1-\eta )}{1+2\delta }|1-\Delta |:}\hfill & |1-\Delta |\ge 1,\hfill \\ {\displaystyle \frac{\mu (1-\eta )}{1+2\delta }:}\hfill & 0\le |1-\Delta |\le 1.\hfill \end{array}\right.$$

(62)

Proof. 

For $m=2$ and $m=3$ in Equations (55) and (56), respectively, we obtain,

$$(1+\delta )b_2=u_1,$$

(63)

$$(1+2\delta )b_3=u_2,$$

(64)

$$-(1+\delta )b_2=v_1$$

(65)

and

$$(1+2\delta )(2b_2^2-b_3)=v_2.$$

(66)

Using Lemma 1 in Equations (63) or (65), we have

$$|b_2|\le {\displaystyle \frac{\mu (1-\eta )}{1+\delta }}.$$

(67)

Adding Equations (64) and (66), we obtain

$$b_2^2={\displaystyle \frac{u_2+v_2}{2(1+2\delta )}}.$$

(68)

Using Lemma 1 in Equation (68), we obtain

$$|b_2^2|\le {\displaystyle \frac{\mu (1-\eta )}{1+2\delta }}.$$

(69)

Hence from Equations (67) and (69), we obtain Equation (60). Using Lemma 1 in Equation (64), we obtain Equation (61). For some real number $\Delta$ and from Equations (64) and (68), we obtain

$$b_3-\Delta b_2^2={\displaystyle \frac{1}{2(1+2\delta )}}\left[\left(\mathsf{\Psi }(\Delta )+1\right)u_2+\left(\mathsf{\Psi }(\Delta )-1\right)v_2\right],$$

(70)

where $\mathsf{\Psi }(\Delta )=1-\Delta .$ Using Lemma 1 in Equation (70), we obtain Equation (62). This completes the proof of Theorem 6. □

Remark 5.

(i) Theorem 6 verifies the bounds of $|b_2|$ and $|b_3|$ obtained by Sharma et al. [12]

(ii) The choice of $\mu =2$ in Theorem 6 verifies the bounds of $|b_2|$ and $|b_3|$ obtained by Jahangiri [22].

Theorem 7.

For $2\le \mu \le 4,$ $0\le \eta <1$ and $\lambda \ge 1$, let $f\in {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )$ and $g\in {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta ).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{\mu (1-\eta )}{m[1+\lambda (m-1\left)\right]}},\forall m\ge 3.$$

(71)

Proof. 

Since $f\in {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta ),$ according to Definition 4, there exists functions $u\left(z\right)$ and $v\left(w\right)$ belonging to ${\mathcal{P}}_{\mu }\left(\eta \right)$ such that

$$f^{\prime }\left(z\right)+\lambda z{f}^{″}\left(z\right)=u\left(z\right)$$

(72)

and

$$g^{\prime }\left(w\right)+\lambda w{g}^{″}\left(w\right)=v\left(w\right),$$

(73)

where $u\left(z\right)$ and $v\left(w\right)$ are given in the form Equations (9) and (10). Therefore, using the Faber polynomial, we obtain

$$f^{\prime }\left(z\right)+\lambda z{f}^{″}\left(z\right)=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(u_1,u_2,\dots ,u_m)z^m$$

(74)

and

$$g^{\prime }\left(w\right)+\lambda w{g}^{″}\left(w\right)=1+\sum _{m=1}^{\infty }{\mathcal{K}}_m^1(v_1,v_2,\dots ,v_m)w^m.$$

(75)

For any function f given by Equation (1), we obtain

$$f^{\prime }\left(z\right)+\lambda z{f}^{″}\left(z\right)=1+\sum _{m=2}^{\infty }[m+\lambda m(m-1)]b_m{z}^{m-1}$$

(76)

and for its analytic continuation map $g=f^{-1},$ we have

$$\begin{array}{cc}\hfill g^{\prime }\left(w\right)+\lambda g^{″}\left(w\right)=& 1+\sum _{m=1}^{\infty }[m+\lambda m(m-1)]d_m{w}^{m-1}\hfill \\ \hfill =& 1+\sum _{m=1}^{\infty }[m+\lambda m(m-1)]{\displaystyle \frac{1}{m}}{\mathcal{K}}_{m-1}^{-m}(b_2,b_3,\dots ,b_m)w^{m-1}.\hfill \end{array}$$

(77)

Hence, from Equations (74) and (76), we obtain

$$[m+\lambda m(m-1)]b_m={\mathcal{K}}_{m-1}^1(u_1,u_2,\dots ,u_{m-1}).$$

(78)

Similarly, from Equations (75) and (78), we obtain

$${\displaystyle \frac{1}{m}}[m+\lambda m(m-1)]{\mathcal{K}}_{m-1}^{-m}(b_0,b_1,\dots ,b_m)={\mathcal{K}}_{m-1}^1(v_1,v_2,\dots ,v_{m-1}).$$

(79)

Now for $b_n=0;$ $2\le n\le m-1,$ we have $d_m=-b_m$ and

$$[m+\lambda m(m-1)]b_m=u_{m-1}$$

(80)

and

$$-[m+\lambda m(m-1)]b_m=v_{m-1}.$$

(81)

Hence, we have

$$b_m={\displaystyle \frac{u_{m-1}}{m+\lambda m(m-1)}}=-{\displaystyle \frac{v_{m-1}}{m+\lambda m(m-1)}}.$$

(82)

Using Lemma 1 in Equation (82), we obtain Equation (71). This completes the proof of Theorem 7. □

For $\mu =2$ in Theorem 7, we obtain the following corollary.

Corollary 5.

For $2\le \mu \le 4,$ $0\le \eta <1$ and $\lambda \ge 1$, let $f\in {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )\equiv {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }\left(\eta \right)$ and $g\in {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )\equiv {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }\left(\eta \right).$ If $b_n=0;$ $2\le n\le m-1,$ then

$$|b_m|\le {\displaystyle \frac{\mu (1-\eta )}{m[1+\lambda (m-1\left)\right]}},\forall m\ge 3.$$

Theorem 8.

For $2\le \mu \le 4$ and $0\le \eta <1$, let $f\in {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta )$ and $g\in {\mathcal{F}}_{\mathsf{\Sigma }}^{\lambda }(\mu ,\eta ).$ Then

$$|b_2|\le \sqrt{{\displaystyle \frac{\mu (1-\eta )}{3(1+2\lambda )}}},$$

(83)

$$|b_3|\le {\displaystyle \frac{\mu (1-\eta )}{3(1+2\lambda )}}$$

(84)

and

$$|b_3-\Delta b_2^2|\le \left\{\begin{array}{cc}{\displaystyle \frac{\mu (1-\eta )}{3(1+2\lambda )}|1-\Delta |:}\hfill & |1-\Delta |\ge 1,\hfill \\ {\displaystyle \frac{\mu (1-\eta )}{3(1+2\lambda )}:}\hfill & 0\le |1-\Delta |\le 1.\hfill \end{array}\right.$$

(85)

The proof of Theorem 8 is much akin to that of Theorem 2, and hence we omit the details involved.

Remark 6.

(i) Theorem 8 verifies the bounds of $|b_2|$ and $|b_3|$ obtained by Sharma et al. [12].

(ii) $\lambda =1$ in Theorem 8 verifies the bound obtained by Sharma et al. [6].

4. Concluding Remarks and Observations

In this article, using the Faber polynomial expansions, we found general coefficient bounds for well-known classes of bi-univalent functions with bounded boundary rotation and bounded radius rotation. We also derived interesting and familiar Fekete–Szegö inequality for these new classes. In addition, we gave interesting remarks and corollaries for certain choices of parameters. Similar coefficient problems may also be analyzed for other function classes existing in the literature. Moreover, several problems that emerge by virtue of convolution, as well as different operators that can be defined, are potentially interesting ones for future researchers and are left for interested readers.