Faber Polynomial Coefficient Estimates of m-Fold Symmetric Bi-Univalent Functions with Bounded Boundary Rotation

Abstract

In the current article, we introduce several new subclasses of m-fold symmetric analytic and bi-univalent functions associated with bounded boundary and bounded radius rotation within the open unit disk $\mathbb{D}.$ Utilizing the Faber polynomial expansion, we derive upper bounds for the coefficients $|b_{mk+1}|$ and establish initial coefficient bounds for $|b_{m+1}|$ and $|b_{2m+1}|.$ Additionally, we explore the Fekete–Szegö inequalities applicable to the functions that fall within these newly defined subclasses.

1. Introduction

Let $\mathcal{S}$ represent the set of functions characterized by the form

$$h(z)=z+\sum _{k=2}^{\infty }{b}_k{z}^k$$

(1)

which are analytic and univalent in the open unit disc $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}.$ Furthermore, each function h possesses an inverse, denoted as $h^{-1},$ which is defined by

$$z=h^{-1}(h(z)),z\in \mathbb{D}$$

and

$$v=h(h^{-1}(v));\left|v\right|<{\displaystyle \frac{1}{4}}.$$

Hence, the inverse function is given by

$$h^{-1}(v)=\xi (v)=v-b_2{v}^2+(2b_2^2-b_3)v^3-(5b_2^3-5b_2{b}_3+b_4)v^4+\cdots .$$

(2)

A function h is considered bi-univalent in the open unit disk $\mathbb{D}$ if both h and its inverse $h^{-1}$ are univalent in $\mathbb{D}.$ Let $\Sigma$ represent the class of bi-univalent functions within $\mathbb{D}.$ For a function f belonging to the class $\Sigma ,$ Lewin [1] demonstrated that $|b_2| < 1.51,$ while Brannan and Cluni [2] established that $|b_2| \le \sqrt{2}.$ Netanyahu [3] determined that the maximum value of $|b_2|$ is $4/3.$ Additionally, Brannan and Taha [4] proposed a specific subclass of bi-univalent functions within the class $\Sigma .$ In recent years, researchers such as Srivastava et al. [5], Frasin and Aouf [6], Altinkaya and Yalcin [7], and Hayami and Owa [8] have explored various subclasses of analytic and bi-univalent functions. Notably, the influential research conducted by Srivastava et al. [9] has reinvigorated the exploration of analytic and bi-univalent functions in recent years, resulting in a significant increase in related publications by Srivastava et al. [9,10,11,12] and other researchers [13,14,15].

A function that is analytic and locally univalent within a simply connected domain is referred to as a function of bounded boundary rotation if the total variation in the direction angle of the tangent to the boundary curve is finite when traversing the boundary in a complete circuit. Paatero [16] defined and examined the class $\mathcal{V}(\vartheta ),$ which encompasses all functions exhibiting boundary rotation not exceeding $\vartheta \pi .$ A function h that belongs to $\mathcal{V}(\vartheta )$ will possess a specific representation,

$$h^{\prime }(z)=exp\left\{-{\int }_0^{2\pi }log|1-z{e}^{-it}|\right\}d\eta (t).$$

Here, $\eta (t)$ is a function with bounded variation that is real-valued and satisfies the conditions

$${\int }_0^{2\pi }d\eta (t)=2and{\int }_0^{2\pi }\left|d\eta (t)\right|\le \vartheta ,\vartheta \ge 2.$$

(3)

The investigation of the class $\mathcal{R}(\vartheta ),$ which encompasses all functions with a rotation radius not exceeding $\vartheta \pi ,$ was initiated and examined by Robertson [17]. Consequently, the function h belonging to $\mathcal{R}(\vartheta )$ can be expressed in an integral form as follows,

$$h(z)=zexp\left\{-{\int }_0^{2\pi }log|1-z{e}^{-it}|\right\}d\eta (t).$$

Here, $\eta (t)$ represents a previously defined function that meets the criteria outlined in (3). The notation $\mathcal{P}(\vartheta )$ refers to the set of functions $\varphi (0)=1,$ which are analytic within the region $\mathbb{D}$ and possess a specific representation,

$$\varphi (z)={\int }_0^{2\pi }\left|{\displaystyle \frac{1+z{e}^{-it}}{1-z{e}^{-it}}}\right|d\eta (t).$$

Here, $\eta (t)$ represents a previously defined function that meets the criteria outlined in (3). This class was first introduced and examined by Pinchuk in 1971.

A domain $\mathcal{D}$ is considered m-fold symmetric if

$$h\left(e^{{\displaystyle \frac{2\pi i}{m}}}z\right)=e^{{\displaystyle \frac{2\pi i}{m}}}h(z).$$

The univalent function $g(z)$ transforms the unit disk $\mathbb{D}$ into a region exhibiting m-fold symmetry and can be expressed as

$$g(z)=\sqrt[m]{h(z^m)}$$

Additionally, g also transforms $\mathbb{D}$ into a region exhibiting m-fold symmetry. A function with 2-fold symmetry is classified as an odd function, as the aforementioned condition simplifies to $h(-z)=-h(z).$

Let ${\mathcal{A}}_m$ represent the set of all m-fold symmetric functions (refer to [18]) provided it possesses the specified normalized form

$$h_m(z)=z+\sum _{k=1}^{\infty }{b}_{km+1}{z}^{km+1}.$$

(4)

The collection of all m-fold symmetric univalent functions is represented by ${\mathcal{S}}_m.$ We define the families of functions as follows: ${\mathcal{S}}_m^{*}$ represents m-fold symmetric starlike functions, ${\mathcal{C}}_m$ denotes convex functions, ${\mathcal{K}}_m$ refers to close-to-convex functions, ${\mathcal{V}}_m(\vartheta )$ indicates functions with a bounded boundary rotation of at most $\vartheta \pi ,$ and ${\mathcal{R}}_m(\vartheta )$ signifies functions with a bounded radius rotation of at most $\vartheta \pi .$

The idea of m-fold symmetric bi-univalent functions was first presented by Srivastava et al. [11]. It has been noted that every function h belonging to $\Sigma$ produces an m-fold symmetric bi-univalent function for every integer $m\in \mathbb{N}.$ The normalized representation of $h_m$ is provided in Equation (4), while $h_m^{-1}$ is expressed as follows. The series expansion for $h_m^{-1}$ is outlined below:

$$\begin{array}{cc}\hfill {\xi }_m(v)=h_m^{-1}(v)=& v-b_{m+1}{v}^{m+1}+\left[(m+1)b_{m+1}^2-b_{2m+1}\right]v^{2m+1}\hfill \\ \hfill -& \left[{\displaystyle \frac{1}{2}}(m+1)(3m+2)b_{m+1}^3-(3m+2)b_{m+1}{b}_{2m+1}+b_{3m+1}\right]v^{3m+1}+\cdots .\hfill \end{array}$$

(5)

We refer to the class of m-fold symmetric bi-univalent functions in $\mathbb{D}$ as ${\Sigma }_m.$ Therefore, when $m=1,$ Formula (5) aligns with Formula (2).

Brannan [19] established that $\mathcal{V}(\vartheta )$ is a subset of the class $\mathcal{K}(\varphi )$ of close-to-convex functions of order $\varphi ={\displaystyle \frac{\vartheta }{2}}-1,$ and Koepf [18] demonstrated that ${\mathcal{V}}_m(\vartheta )$ is a subset of the class ${\mathcal{K}}_m(\varphi )$ of m-fold symmetric close-to-convex functions of order $\varphi ={\displaystyle \frac{\vartheta -2}{2m}}.$ This results in addressing the coefficient issue for m-fold symmetric functions with restricted boundary rotation when $\vartheta \ge 2m.$ Additionally, for $m\in \mathbb{N},$${\mathcal{V}}_m(2m+2)$ consists of close-to-convex functions and hence are univalent functions. Leach [20] also explored the notion of odd univalent functions characterized by limited boundary rotation, and later that year, he expanded this concept to include m-fold symmetric functions. Leach [21] demonstrated that if $h_m$ belongs to ${\mathcal{V}}_m(\vartheta ),$ then

$$|b_{m+1}| \le {\displaystyle \frac{\vartheta }{m(m+1)}},\vartheta \ge 2m$$

and

$$|b_{2m+1}| \le \left\{\begin{array}{cc}{\displaystyle \frac{{\vartheta }^2+m}{2m^2(m+1)}}\hfill & :\vartheta \ge 2m,\hfill \\ {\displaystyle \frac{4m\vartheta +6\vartheta +4}{2m(2m+1)(4m+2-\vartheta )}}\hfill & :2\le \vartheta <2m.\hfill \end{array}\right.$$

In [22], Faber presented Faber polynomials, which were subsequently examined by Gong [23] within the context of geometric function theory. In their published research, certain advancements have been achieved in determining the general coefficient bounds $|b_k|$ through the application of Faber polynomial expansions. However, there has been minimal exploration regarding the coefficient bounds $|b_k|$ for $k\ge 4$ within Maclaurin’s series using these expansions. For further investigation, refer to sources [24,25]. The coefficients of the inverse map $\xi =h^{-1}$ can be expressed using the Faber polynomial expansion for functions $h\in \mathcal{A}$ of the form (1), as indicated in [25]. Let us assume that

$$\begin{array}{cc}\hfill \xi (v)=h^{-1}(v)& =v+\sum _{k=2}^{\infty }{B}_k{v}^k\hfill \\ \hfill & =v+\sum _{k=2}^{\infty }{\displaystyle \frac{1}{k}}{\mathcal{Q}}_{k-1}^{-k}(b_2,b_3,\dots ,b_k)v^k,\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill {\mathcal{Q}}_{k-1}^{-k}& ={\displaystyle \frac{(-k)!}{(1-2k)!(k-1)!}}{b}_2^{k-1}+{\displaystyle \frac{(-k)!}{\left(2(1-k)\right)!(-3+k)!}}{b}_2^{k-3}{b}_3\hfill \\ \hfill & +{\displaystyle \frac{(-k)!}{(3-2k)!(-4+k)!}}{b}_2^{k-4}{b}_4\hfill \\ \hfill & +{\displaystyle \frac{(-k)!}{\left(2(2-k)\right)!(-5+k)!}}{b}_2^{k-5}\left[b_5+(2-k)b_3^2\right]\hfill \\ & +{\displaystyle \frac{(-k)!}{(5-2k)!(k-6)!}}{b}_2^{k-6}\left[b_6+(5-2k)b_2{b}_4\right]+\sum _{j=1}^{\infty }{b}^{k-j}{\mathcal{W}}_j,\hfill \end{array}$$

and ${\mathcal{W}}_j$ for $j\in [7,m]$ in a homogeneous polynomial in the variable $b_2,b_3,\dots ,b_k$ and expressions such as (for example) $(-k)!$ are symbolically interpreted as follows:

$$(-k)!\equiv \Gamma (1-n):=(-n)(-n-1)(-n-2)\cdots (k\in \mathbb{N}\cup \{0\}).$$

The first three terms of ${\mathcal{Q}}_{k-1}^{-k}$ are

$${\mathcal{Q}}_1^{-2}=-2b_2,$$

$${\mathcal{Q}}_2^{-3}=3(2b_2^2-b_3),$$

and

$${\mathcal{Q}}_3^{-4}=-4(5b_2^3-5b_2{b}_3+b_4).$$

In general, an expansion of ${\mathcal{B}}_k^q$ is

$${\mathcal{B}}_k^q=q{b}_k+{\displaystyle \frac{q(q-1)}{2}}{\mathcal{D}}_k^2+{\displaystyle \frac{q!}{(q-3)!3!}}{\mathcal{D}}_k^3+\dots +{\displaystyle \frac{q!}{(q-k)!k!}}{\mathcal{D}}_k^k,$$

where ${\mathcal{B}}_k^q={\mathcal{B}}_k^q(b_1,b_3,\dots ,b_k)$ and

$${\mathcal{D}}_k^q(b_1,b_2,\dots ,b_k)=\sum _{k=1}^{\infty }{\displaystyle \frac{(k)!b_1^{{\gamma }_1}{b}_2^{{\gamma }_2}\dots b_k^{{\gamma }_k}}{{\gamma }_1!{\gamma }_2!\dots {\gamma }_k!}},$$

where $a_1=1$ and the sum is taken over all non-negative integers ${\gamma }_1!,{\gamma }_2!,\dots {\gamma }_k!$ such that

$$\begin{array}{cc}\hfill {\gamma }_1+{\gamma }_2+\dots +{\gamma }_k& =n\hfill \\ \hfill {\gamma }_1+2{\gamma }_2+\dots +k{\gamma }_k& =k.\hfill \end{array}$$

In a similar manner, by using the Faber polynomial expansion for functions $h_m\in {\mathcal{A}}_m$ of the form (4), that is,

$$h_m(u)=u+\sum _{k=1}^{\infty }{\mathcal{Q}}_k^{{\displaystyle \frac{1}{m}}}(b_2,b_3,\cdots ,b_{k+1})u^{mk+1}.$$

The coefficients of the inverse map ${\xi }_m(v)=h_m^{-1}$ can be represented as follows:

$${\xi }_m(v)=v+\sum _{k=1}^{\infty }{\displaystyle \frac{1}{mk+1}}{\mathcal{Q}}_k^{-(mk+1)}(b_{m+1},b_{2m+1},\cdots ,b_{km+1})v^{mk+1}.$$

Let $h_m\in {\mathcal{S}}_m$ of the form (4), then

$${\displaystyle \frac{u{h}_m^{\prime }(u)}{h_m(u)}}=1-\sum _{k=1}^{\infty }{\mathcal{X}}_{mk}(b_{m+1},b_{2m+1},\cdots ,b_{km+1})u^{mk},$$

where

$$\begin{array}{cc}\hfill {\mathcal{X}}_{mk}(b_{m+1},b_{2m+1},\cdots ,b_{km+1})=& {\mathcal{F}}_{mk}(0,\cdots ,0,b_{m+1},0,\cdots ,0,b_{mk+1})\hfill \\ \hfill =& \sum _{m{j}_m+2m{j}_{2m}+\cdots +mk{j}_{mk}}{\mathcal{Y}}_{(j_1,j_2,\cdots ,j_{mk})}{b}_{m+1}^{j_m}{b}_{2m+1}^{j_{2m}}\cdots b_{mk+1}^{j_{mk}}.\hfill \end{array}$$

In the current article, we introduce several new subclasses of m-fold symmetric analytic and bi-univalent functions associated with bounded boundary and bounded radius rotation within the open unit disk $\mathbb{D}.$ Utilizing the Faber polynomial expansion, we derive upper bounds for the coefficients $|b_{mk+1}|$ and establish initial coefficient bounds for $|b_{m+1}|$ and $|b_{2m+1}|.$ Additionally, we explore the Fekete–Szegö inequalities applicable to the functions that fall within these newly defined subclasses.

2. Main Definitions

Let $\vartheta \ge 2$ and $0\le \lambda <1.$ Define ${\mathcal{P}}_{\vartheta }(\lambda )$ as the set of functions $\psi (u)$ that are analytic and normalized such that $\psi (0)=1,$ and that meet the specified condition

$${\int }_0^{2\pi }\left|{\displaystyle \frac{\Re (\psi (u))-\lambda }{1-\lambda }}\right| dt\le \vartheta \pi .$$

where $u=r{e}^{it}\in \mathbb{D}.$ The class ${\mathcal{P}}_{\vartheta }(\lambda )$ was established by Padmanabhan and Parvatham [26].

Definition 1.

Assume that $m\ge 1,$$0\le \lambda <1$ and $\vartheta \ge 2m.$ A function $h_m$ belonging to ${\Sigma }_m,$ as defined in (4), is considered to be in the class ${\mathcal{H}}_{\Sigma }^m(\vartheta ,\lambda ),$ if both conditions

$$h_m^{\prime }(z)\in {\mathcal{P}}_{\vartheta }(\lambda )$$

and

$${\xi }_m^{\prime }(v)\in {\mathcal{P}}_{\vartheta }(\lambda )$$

are met, where ${\xi }_m$ represents the analytic continuation of $h_m^{-1}$ as specified in (5).

Remark 1. (i) Selecting $m=1$ in Definition 1 results in the class ${\mathcal{H}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{H}}_{\Sigma }(\vartheta ,\lambda ),$ as introduced and examined by Sharma et al. [27].

(ii) Selecting $m=1$ and $\vartheta =2$ in Definition 1 results in the class ${\mathcal{H}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{H}}_{\Sigma }(\lambda ),$ as introduced and examined by Srivastava et al. [12].

(iii) Selecting $\vartheta =2$ in Definition 1 results in the class ${\mathcal{H}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{H}}_{\Sigma }^m(\lambda ),$ as introduced and examined by Srivastava et al. [11].

Definition 2.

Assume that $m\ge 1,$$0\le \lambda <1$ and $\vartheta \ge 2m.$ A function $h_m$ belonging to ${\Sigma }_m,$ as defined in (4), is considered to be in the class ${\mathcal{R}}_{\Sigma }^m(\vartheta ,\lambda ),$ if both conditions

$${\displaystyle \frac{u{h}_m^{\prime }(z)}{h_m(z)}}\in {\mathcal{P}}_{\vartheta }(\lambda )$$

and

$${\displaystyle \frac{v{\xi }_m^{\prime }(v)}{{\xi }_m(v)}}\in {\mathcal{P}}_{\vartheta }(\lambda )$$

are met, where ${\xi }_m$ represents the analytic continuation of $h_m^{-1}$ as specified in (5).

Remark 2. (i) Selecting $m=1$ in Definition 2 results in the class ${\mathcal{R}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{R}}_{\Sigma }(\vartheta ,\lambda ),$ as introduced by Li et al. [28] and examined by Sharma et al. [27].

(ii) Selecting $m=1$ and $\vartheta =2$ in Definition 2 results in the class ${\mathcal{R}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{S}}_{\Sigma }^{*}(\lambda ),$ as introduced and examined by Brannan and Taha [4].

(iii) Selecting $\vartheta =2$ in Definition 2 results in the class ${\mathcal{R}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{S}}_{\Sigma }^{*,m}(\lambda ),$ as introduced and examined by Sivasubramanian et al. [10].

Definition 3.

Assume that $m\ge 1,$$0\le \lambda <1$, and $\vartheta \ge 2m.$ A function $h_m$ belonging to ${\Sigma }_m,$ as defined in (4), is considered to be in the class ${\mathcal{V}}_{\Sigma }^m(\vartheta ,\lambda ),$ if both conditions

$$1+{\displaystyle \frac{u{h}_m^{″}(z)}{h_m^{\prime }(z)}}\in {\mathcal{P}}_{\vartheta }(\lambda )$$

and

$$1+{\displaystyle \frac{v{\xi }_m^{″}(v)}{{\xi }_m^{\prime }(v)}}\in {\mathcal{P}}_{\vartheta }(\lambda )$$

are met, where ${\xi }_m$ represents the analytic continuation of $h_m^{-1}$ as specified in (5).

Remark 3. (i) Selecting $m=1$ in Definition 3 results in the class ${\mathcal{V}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{V}}_{\Sigma }(\vartheta ,\lambda ),$ as introduced by Li et al. [28] and examined by Sharma et al. [27].

(ii) Selecting $m=1$ and $\vartheta =2$ in Definition 3 results in the class ${\mathcal{V}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{C}}_{\Sigma }(\lambda ),$ as introduced and examined by Brannan and Taha [4].

(iii) Selecting $\vartheta =2$ in Definition 3 results in the class ${\mathcal{V}}_{\Sigma }^m(\vartheta ,\lambda )$ being reduced to the class ${\mathcal{C}}_{\Sigma }^m(\lambda ),$ as introduced and examined by Sivasubramanian et al. [10].

In order to demonstrate the theorems presented in the main sections, we require the following lemma, which is stated as follows.

Lemma 1

([29]). Let $\psi (z)$ be defined as $\psi (z)=1+{\sum }_{k=1}^{\infty }{\psi }_k{z}^k,$ where $z\in \mathbb{D}$ and ψ belongs to ${\mathcal{P}}_{\vartheta }(\lambda ),$ then

$$|{\psi }_k| \le \vartheta (1-\lambda ),k\in \mathbb{N}.$$

Assume that the functions $x_m(z)$ and $y_m(v)$ belong to the class ${\mathcal{P}}_{\vartheta }(\lambda )$ and can be defined as:

$$x_m(u)=1+x_m{z}^m+x_{2m}{z}^{2m}+x_{3m}{z}^{3m}+\cdots$$

(6)

and

$$y_m(v)=1+y_m{v}^m+y_{2m}{v}^{2m}+y_{3m}{v}^{3m}+\cdots$$

(7)

It follows from Lemma 1, that

$$|x_{km}| \le \vartheta (1-\lambda ),k,m\in \mathbb{N}$$

(8)

and

$$|y_{km}| \le \vartheta (1-\lambda ),k,m\in \mathbb{N}.$$

(9)

3. Main Results

Theorem 1.

Assume that $m\ge 1,$$0\le \lambda <1$, and $\vartheta \ge 2m.$ A function $h_m\in {\mathcal{H}}_{\Sigma }^m(\vartheta ,\lambda ),$ and if $b_{mn+1}=0,$$1\le n\le k-1,$ then

$$|b_{mk+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )}{mk+1}},\forall k\ge 2.$$

(10)

Proof. 

According to Definition 1, a function $h_m\in {\mathcal{H}}_{\Sigma }^m(\vartheta ,\lambda )$; then, there exist two functions $x(x)$ and $y(v)$ that belong to the family ${\mathcal{P}}_{\vartheta }(\lambda ),$ and we have

$$h_m^{\prime }(z)=x_m(z)$$

(11)

and

$${\xi }_m^{\prime }(v)=y_m(v),$$

(12)

where $x(z)$ and $y(v)$ are represented in the form (6) and (7), respectively. For the function $h_m$ represented in the form (4), we have

$$h_m^{\prime }(z)=1+\sum _{k=1}^{\infty }(mk+1)b_{mk+1}{z}^{mk}.$$

(13)

For its inverse map ${\xi }_m=h_m^{-1},$ we obtain

$$\begin{array}{cc}\hfill {\xi }_m^{\prime }(v)=& 1+\sum _{k=1}^{\infty }(mk+1)B_{mk+1}{v}^{mk}\hfill \\ \hfill =& 1+\sum _{k=1}^{\infty }{\mathcal{Q}}_k^{-(mk+1)}(b_{m+1},b_{2m+1},\cdots ,b_{km+1})v^{mk}.\hfill \end{array}$$

(14)

On the other hand,

$$h_m^{\prime }(z)=1+\sum _{k=1}^{\infty }{\mathcal{D}}_k^1(x_m,x_{2m},\cdots ,x_{km})z^{mk}$$

(15)

and

$${\xi }_m^{\prime }(v)=1+\sum _{k=1}^{\infty }{\mathcal{D}}_k^1(y_m,y_{2m},\cdots ,y_{km})v^{mk}.$$

(16)

By examining the coefficients of Equations (13) and (15), we find that

$$(mk+1)b_{mk+1}={\mathcal{D}}_k^1(x_m,x_{2m},\cdots ,x_{km}).$$

(17)

Similarly, by examining the coefficients of Equations (14) and (16), we find that

$${\mathcal{Q}}_k^{-(mk+1)}(b_{m+1},b_{2m+1},\cdots ,b_{km+1})={\mathcal{D}}_k^1(y_m,y_{2m},\cdots ,y_{km}).$$

(18)

For $b_{mn+1}=0,$$1\le n\le k-1,$ we have $B_{mk+1}=a_{mk+1}=0,$

$$(mk+1)b_{mk+1}=x_{mk}$$

(19)

and

$$-(mk+1)b_{mk+1}=y_{mk}.$$

(20)

By taking the absolute values of Equations (19) and (20), and utilizing Equations (8) and (9), we obtain

$$|b_{mk+1}| \le {\displaystyle \frac{|x_{mk}|}{mk+1}}={\displaystyle \frac{|y_{mk}|}{mk+1}}\le {\displaystyle \frac{\vartheta (1-\lambda )}{mk+1}}.$$

(21)

This concludes the proof of Theorem 1. □

Theorem 2.

Assume that $m\ge 1,$$0\le \lambda <1$, and $\vartheta \ge 2m.$ A function $h_m\in {\mathcal{H}}_{\Sigma }^m(\vartheta ,\lambda ),$ then

$$|b_{m+1}| \le \sqrt{{\displaystyle \frac{2\vartheta (1-\lambda )}{(m+1)(2m+1)}}},$$

(22)

$$|b_{2m+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )}{2m+1}}$$

(23)

and

$$|b_{2m+1}-\delta b_{m+1}^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\lambda )(m+1-2\delta )}{(m+1)(2m+1)}}\hfill & \mathit{if}\delta <0,\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )}{(m+1)(2m+1)}}\hfill & \mathit{if}0\le \delta \le 2,\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )(2\delta -1-m)}{(m+1)(2m+1)}}\hfill & \mathit{if}\delta >2,\hfill \end{array}\right.$$

(24)

where δ represents a real number.

Proof. 

By substituting $k=2$ in Equations (17) and (18), respectively, we obtain

$$(2m+1)b_{2m+1}=x_{2m}$$

(25)

and

$$(2m+1)(m+1)b_{m+1}^2-(2m+1)b_{2m+1}=y_{2m}.$$

(26)

By summing (25) and (26), we obtain

$$(2m+1)(m+1)b_{m+1}^2=x_{2m}+y_{2m}.$$

(27)

By taking the absolute values of Equation (27), and utilizing Equations (8) and (9), we obtain the bound of $|b_{m+1}|$ given in (22). Similarly, by taking the absolute values of Equation (25), and utilizing Equations (8) and (9), we obtain the bound of $|b_{2m+1}|$ given in (23). For any $\delta$ in $\mathbb{R}$ and based on Equations (25) and (27), we obtain

$$b_{2m+1}-\delta b_{m+1}^2={\displaystyle \frac{[1+m-\delta ]x_{2m}-\delta y_{2m}}{(m+1)(2m+1)}}.$$

(28)

By taking the absolute values of Equation (28), and utilizing Equations (8) and (9), we obtain the bound of $|b_{2m+1}-\delta b_{m+1}^2|$ given in (24). This concludes the proof of Theorem 2. □

Theorem 3.

Assume that $m\ge 1,$$0\le \lambda <1$, and $\vartheta \ge 2m.$ A function $h_m\in {\mathcal{R}}_{\Sigma }^m(\vartheta ,\lambda ),$ and if $b_{mn+1}=0,$$1\le n\le k-1,$ then

$$|b_{mk+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )}{mk}},\forall k\ge 2.$$

(29)

Proof. 

According to Definition 2, a function $h_m\in {\mathcal{R}}_{\Sigma }^m(\vartheta ,\lambda )$; then, there exiss two functions $x(z)$ and $y(v)$ belonging to the family ${\mathcal{P}}_{\vartheta }(\lambda ),$ and we have

$${\displaystyle \frac{z{h}_m^{\prime }(z)}{h_m(z)}}=x_m(z)$$

(30)

and

$${\displaystyle \frac{v{\xi }_m^{\prime }(v)}{{\xi }_m(v)}}=y_m(v),$$

(31)

where $x(z)$ and $y(v)$ are represented in the form (6) and (7), respectively. For the function $h_m$ represented in the form (4), we have

$${\displaystyle \frac{z{h}_m^{\prime }(z)}{h_m(z)}}=1-\sum _{k=1}^{\infty }{\mathcal{X}}_{mk}(b_{m+1},b_{2m+1},\cdots ,b_{km+1})z^{mk}.$$

(32)

For its inverse map ${\xi }_m=h_m^{-1},$ we obtain

$${\displaystyle \frac{v{\xi }_m^{\prime }(v)}{{\xi }_m(v)}}=1-\sum _{k=1}^{\infty }{\mathcal{X}}_{mk}(B_{m+1},B_{2m+1},\cdots ,B_{km+1})v^{mk},$$

(33)

where

$$B_{mk+1}={\displaystyle \frac{1}{mk+1}}{\mathcal{Q}}_k^{-(mk+1)}(b_{m+1},b_{2m+1},\cdots ,b_{km+1}).$$

On the other hand,

$${\displaystyle \frac{z{h}_m^{\prime }(z)}{h_m(z)}}=1+\sum _{k=1}^{\infty }{\mathcal{D}}_k^1(x_m,x_{2m},\cdots ,x_{km})z^{mk}$$

(34)

and

$${\displaystyle \frac{v{\xi }_m^{\prime }(v)}{{\xi }_m(v)}}=1+\sum _{k=1}^{\infty }{\mathcal{D}}_k^1(y_m,y_{2m},\cdots ,y_{km})v^{mk}.$$

(35)

By examining the coefficients of Equations (32) and (34), we find that

$$-{\mathcal{X}}_{mk}(b_{m+1},b_{2m+1},\cdots ,b_{km+1})={\mathcal{D}}_k^1(x_m,x_{2m},\cdots ,x_{km}).$$

(36)

Similarly, by examining the coefficients of Equations (33) and (35), we find that

$$-{\mathcal{X}}_{mk}(B_{m+1},B_{2m+1},\cdots ,B_{km+1})={\mathcal{D}}_k^1(y_m,y_{2m},\cdots ,y_{km}).$$

(37)

For $b_{mn+1}=0,$$1\le n\le k-1,$ we have $B_{mk+1}=a_{mk+1}=0,$

$$mk{b}_{mk+1}=x_{mk},$$

(38)

and

$$-mk{b}_{mk+1}=y_{mk}.$$

(39)

By taking the absolute values of Equations (38) and (39), and utilizing Equations (8) and (9), we obtain

$$|b_{mk+1}| \le {\displaystyle \frac{|x_{mk}|}{mk}}={\displaystyle \frac{|y_{mk}|}{mk}}\le {\displaystyle \frac{\vartheta (1-\lambda )}{mk}}.$$

(40)

This concludes the proof of Theorem 3. □

Theorem 4.

Assume that $m\ge 1,$$0\le \lambda <1$, and $\vartheta \ge 2m.$ A function $h_m\in {\mathcal{R}}_{\Sigma }^m(\vartheta ,\lambda ),$ then

$$|b_{m+1}| \le {\displaystyle \frac{\sqrt{\vartheta (1-\lambda )}}{m}},$$

(41)

$$|b_{2m+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )(m+1)}{2m^2}}$$

(42)

and

$$|b_{2m+1}-\delta b_{m+1}| \le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\lambda )(m+1-2\delta )}{m^2}}\hfill & {\displaystyle \mathit{if}\delta <\frac{1}{2},}\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )}{2m}}\hfill & {\displaystyle \mathit{if}\frac{1}{2}\le \delta \le m+\frac{1}{2},}\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )(2\delta -1-m)}{2m^2}}\hfill & {\displaystyle \mathit{if}\delta >m+\frac{1}{2},}\hfill \end{array}\right.$$

(43)

where δ represents a real number.

Proof. 

By substituting $k=2$ in Equations (36) and (37), respectively, we obtain

$$2m{b}_{2m+1}-m{b}_{m+1}^2=x_{2m}$$

(44)

and

$$(2m+1)m{b}_{m+1}^2-2m{b}_{2m+1}=y_{2m}.$$

(45)

By summing (44) and (45), we obtain

$$2m^2{b}_{m+1}^2=x_{2m}+y_{2m}.$$

(46)

By taking the absolute values of Equation (46), and utilizing Equations (8) and (9), we obtain the bound of $|b_{m+1}|$ given in (41). Similarly, we can derive the following from Equations (44)–(46), and once more we obtain

$$4m^2{b}_{2m+1}=(2m+1)x_{2m}+y_{2m}.$$

(47)

By taking the absolute values of Equation (47), and utilizing Equations (8) and (9), we obtain the bound of $|b_{2m+1}|$ given in (42). For any $\delta$ in $\mathbb{R}$ and based on Equations (46) and (47), we obtain

$$b_{2m+1}-\delta b_{m+1}^2={\displaystyle \frac{(2m+1-2\delta )x_{2m}+(1-2\delta )y_{2m}}{4m^2}}.$$

(48)

By taking the absolute values of Equation (48), and utilizing Equations (8) and (9), we obtain the bound of $|b_{2m+1}-\delta b_{m+1}^2|$ given in (43). This concludes the proof of Theorem 4. □

Theorem 5.

Assume that $m\ge 1,$$0\le \lambda <1$, and $\vartheta \ge 2m.$ A function $h_m\in {\mathcal{V}}_{\Sigma }^m(\vartheta ,\lambda ),$ and if $b_{mn+1}=0,$$1\le n\le k-1,$ then

$$|b_{mk+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )}{mk(mk+1)}},\forall k\ge 2.$$

(49)

Proof. 

According to Definition 3, a function $h_m\in {\mathcal{V}}_{\Sigma }^m(\vartheta ,\lambda )$; then, there exist two functions $x(z)$ and $y(v)$ belonging to the family ${\mathcal{P}}_{\vartheta }(\lambda ),$ and we have

$$1+{\displaystyle \frac{z{h}_m^{″}(z)}{h_m^{\prime }(z)}}=x_m(z)$$

(50)

and

$$1+{\displaystyle \frac{v{\xi }_m^{″}(v)}{{\xi }_m^{\prime }(v)}}=y_m(v),$$

(51)

where $x(z)$ and $y(v)$ are represented in the form (6) and (7), respectively. For the function $h_m$ represented in the form (4), we have

$$1+{\displaystyle \frac{z{h}_m^{″}(z)}{h_m^{\prime }(z)}}=1-\sum _{k=1}^{\infty }{\mathcal{X}}_{mk}((m+1)b_{m+1},(2m+1)b_{2m+1},\cdots ,(mk+1)b_{km+1})z^{mk}.$$

(52)

For its inverse map ${\xi }_m=h_m^{-1},$ we obtain

$$1+{\displaystyle \frac{v{\xi }_m^{″}(v)}{{\xi }_m^{\prime }(v)}}=1-\sum _{k=1}^{\infty }{\mathcal{X}}_{mk}((m+1)B_{m+1},(m+1)B_{2m+1},\cdots ,(mk+1)B_{km+1})v^{mk},$$

(53)

where

$$B_{mk+1}={\displaystyle \frac{1}{mk+1}}{\mathcal{Q}}_k^{-(mk+1)}((m+1)b_{m+1},(2m+1)b_{2m+1},\cdots ,(km+1)b_{km+1}).$$

On the other hand,

$$1+{\displaystyle \frac{z{h}_m^{″}(z)}{h_m^{\prime }(z)}}=1+\sum _{k=1}^{\infty }{\mathcal{D}}_k^1(x_m,x_{2m},\cdots ,x_{km})z^{mk}$$

(54)

and

$$1+{\displaystyle \frac{v{\xi }_m^{″}(v)}{{\xi }_m^{\prime }(v)}}=1+\sum _{k=1}^{\infty }{\mathcal{D}}_k^1(y_m,y_{2m},\cdots ,y_{km})v^{mk}.$$

(55)

By examining the coefficients of Equations (52) and (54), we find that

$$-{\mathcal{X}}_{mk}((m+1)b_{m+1},(2m+1)b_{2m+1},\cdots ,(mk+1)b_{km+1})={\mathcal{D}}_k^1(x_m,x_{2m},\cdots ,x_{km}).$$

(56)

Similarly, by examining the coefficients of Equations (53) and (55), we find that

$$-{\mathcal{X}}_{mk}((m+1)B_{m+1},(m+1)B_{2m+1},\cdots ,(mk+1)B_{km+1})={\mathcal{D}}_k^1(y_m,y_{2m},\cdots ,y_{km}).$$

(57)

For $b_{mn+1}=0,$$1\le n\le k-1,$ we have $B_{mk+1}=a_{mk+1}=0,$

$$mk(mk+1)b_{mk+1}=x_{mk},$$

(58)

and

$$-mk(mk+1)b_{mk+1}=y_{mk}.$$

(59)

By taking the absolute values of Equations (58) and (59), and utilizing Equations (8) and (9), we obtain

$$|b_{mk+1}| \le {\displaystyle \frac{|x_{mk}|}{mk(mk+1)}}={\displaystyle \frac{|y_{mk}|}{mk(mk+1)}}\le {\displaystyle \frac{\vartheta (1-\lambda )}{mk(mk+1)}}.$$

(60)

This concludes the proof of Theorem 5. □

Theorem 6.

Assume that $m\ge 1,$$0\le \lambda <1$, and $\vartheta \ge 2m.$ A function $h_m\in {\mathcal{V}}_{\Sigma }^m(\vartheta ,\lambda ),$ then

$$|b_{m+1}| \le \sqrt{{\displaystyle \frac{\vartheta (1-\lambda )}{m^2(m+1)}}},$$

(61)

$$|b_{2m+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )}{2m^2}}$$

(62)

and

$$|b_{2m+1}-\delta b_{m+1}^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\lambda )(m+1-2\delta )}{2m^2(m+1)}}\hfill & {\displaystyle \mathit{if}\delta <\frac{{(m+1)}^2}{2(2m+1)},}\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )}{2(2m+1)}}\hfill & {\displaystyle \mathit{if}\frac{{(m+1)}^2}{2(2m+1)}\le \delta \le \frac{(m+1)(3m+1)}{2(2m+1)},}\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )(2\delta -1-m)}{2m^2(m+1)}}\hfill & {\displaystyle \mathit{if}\delta >\frac{(m+1)(3m+1)}{2(2m+1)},}\hfill \end{array}\right.$$

(63)

where δ represents a real number.

Proof. 

By substituting $k=2$ in Equations (56) and (57), respectively, we obtain

$$2m(2m+1)b_{2m+1}-m{(m+1)}^2{b}_{m+1}^2=x_{2m}$$

(64)

and

$$m(m+1)(3m+1)b_{m+1}^2-2m(2m+1)b_{2m+1}=y_{2m}.$$

(65)

By summing (64) and (65), we obtain

$$2m^2(m+1)b_{m+1}^2=x_{2m}+y_{2m}.$$

(66)

By taking the absolute values of Equation (66), and utilizing Equations (8) and (9), we obtain the bound of $|b_{m+1}|$ given in (61). Similarly, we can derive the following from Equations (64)–(66), and once more, we obtain

$$4m^2(2m+1)b_{2m+1}=(3m+1)x_{2m}+(m+1)y_{2m}.$$

(67)

By taking the absolute values of Equation (67), and utilizing Equations (8) and (9), we obtain the bound of $|b_{2m+1}|$ given in (62). For any $\delta$ in $\mathbb{R}$ and based on Equations (66) and (67), we obtain

$$b_{2m+1}-\delta b_{m+1}^2={\displaystyle \frac{[(m+1)(3m+1)-(4m+2)\delta ]x_{2m}+[{(m+1)}^2-(4m+2)\delta ]y_{2m}}{4m^2(m+1)(2m+1)}}.$$

(68)

By taking the absolute values of Equation (68), and utilizing Equations (8) and (9), we obtain the bound of $|b_{2m+1}-\delta b_{m+1}^2|$ given in (63). This concludes the proof of Theorem 6. □

4. Corollaries and Consequences

For $m=1,$ we derive the subsequent corollary from Theorem 1.

Corollary 1.

Assume that $0\le \lambda <1$ and $\vartheta \ge 2m.$ A function $h\in {\mathcal{H}}_{\Sigma }(\vartheta ,\lambda ),$ and if $b_{n+1}=0,$$1\le n\le k-1,$ then

$$|b_k| \le {\displaystyle \frac{\vartheta (1-\lambda )}{k}},\forall k\ge 3.$$

For the case where $m=1$ and $\vartheta =2,$ we derive the following corollary from Theorem 1.

Corollary 2.

Assume that $0\le \lambda <1.$ A function $h\in {\mathcal{H}}_{\Sigma }(\lambda ),$ and if $b_{n+1}=0,$$1\le n\le k-1,$ then

$$|b_k| \le {\displaystyle \frac{2(1-\lambda )}{k}},\forall k\ge 3.$$

For $\vartheta =2,$ we derive the subsequent corollary from Theorem 1.

Corollary 3.

Assume that $m\ge 1$ and $0\le \lambda <1.$ A function $h_m\in {\mathcal{H}}_{\Sigma }^m(\lambda ),$ and if $b_{mn+1}=0,$$1\le n\le k-1,$ then

$$|b_{mk+1}| \le {\displaystyle \frac{2(1-\lambda )}{mk+1}},\forall k\ge 2.$$

For $m=1,$ we derive the subsequent corollary from Theorem 2.

Corollary 4.

Assume that $0\le \lambda <1$ and $\vartheta \ge 2m.$ A function $h\in {\mathcal{H}}_{\Sigma }(\vartheta ,\lambda ),$ then

$$|b_2| \le \sqrt{{\displaystyle \frac{\vartheta (1-\lambda )}{3}}},$$

$$|b_3| \le {\displaystyle \frac{\vartheta (1-\lambda )}{3}}$$

and

$$|b_3-\delta b_{m+1}^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\lambda )(1-\delta )}{3}}\hfill & \mathit{if}\delta <0,\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )}{6}}\hfill & \mathit{if}0\le \delta \le 2,\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )(\delta -1)}{3}}\hfill & \mathit{if}\delta >2,\hfill \end{array}\right.$$

where δ represents a real number.

For $\vartheta =2,$ we derive the subsequent corollary from Theorem 2.

Corollary 5.

Assume that $m\ge 1$ and $0\le \lambda <1.$ A function $h_m\in {\mathcal{H}}_{\Sigma }^m(\lambda ),$ then

$$|b_{m+1}| \le 2\sqrt{{\displaystyle \frac{(1-\lambda )}{(m+1)(2m+1)}}},$$

$$|b_{2m+1}| \le {\displaystyle \frac{2(1-\lambda )}{2m+1}},$$

and

$$|b_{2m+1}-\delta b_{m+1}^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{2(1-\lambda )(m+1-2\delta )}{(m+1)(2m+1)}}\hfill & \mathit{if}\delta <0,\hfill \\ {\displaystyle \frac{2(1-\lambda )}{(m+1)(2m+1)}}\hfill & \mathit{if}0\le \delta \le 2,\hfill \\ {\displaystyle \frac{2(1-\lambda )(2\delta -1-m)}{(m+1)(2m+1)}}\hfill & \mathit{if}\delta >2,\hfill \end{array}\right.$$

where δ represents a real number.

For $m=1,$ we derive the subsequent corollary from Theorem 3.

Corollary 6.

Assume that $0\le \lambda <1$ and $\vartheta \ge 2m.$ A function $h\in {\mathcal{R}}_{\Sigma }(\vartheta ,\lambda ),$ and if $b_{n+1}=0,$$1\le n\le k-1,$ then

$$|b_{k+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )}{k}},\forall k\ge 2.$$

For the case where $m=1$ and $\vartheta =2,$ we derive the following corollary from Theorem 3.

Corollary 7.

Assume that $0\le \lambda <1.$ A function $h\in {\mathcal{S}}_{\Sigma }^{*}(\lambda ),$ and if $b_{n+1}=0,$$1\le n\le k-1,$ then

$$|b_{k+1}| \le {\displaystyle \frac{2(1-\lambda )}{k}},\forall k\ge 2.$$

For $\vartheta =2,$ we derive the subsequent corollary from Theorem 3.

Corollary 8.

Assume that $m\ge 1$ and $0\le \lambda <1.$ A function $h_m\in {\mathcal{R}}_{\Sigma }^{*,m}(\lambda ),$ and if $b_{mn+1}=0,$$1\le n\le k-1,$ then

$$|b_{mk+1}| \le {\displaystyle \frac{2(1-\lambda )}{mk}},\forall k\ge 2.$$

For $m=1,$ we derive the subsequent corollary from Theorem 4.

Corollary 9.

Assume that $0\le \lambda <1$ and $\vartheta \ge 2.$ A function $h\in {\mathcal{R}}_{\Sigma }(\vartheta ,\lambda ),$ then

$$|b_2| \le \sqrt{\vartheta (1-\lambda )},$$

$$|b_3| \le \vartheta (1-\lambda )$$

and

$$|b_3-\delta b_2| \le \left\{\begin{array}{cc}{\displaystyle 2\vartheta (1-\lambda )(1-\delta )}\hfill & {\displaystyle \mathit{if}\delta <\frac{1}{2},}\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )}{2}}\hfill & {\displaystyle \mathit{if}\frac{1}{2}\le \delta \le \frac{3}{2},}\hfill \\ {\displaystyle 2\vartheta (1-\lambda )(\delta -1)}\hfill & {\displaystyle \mathit{if}\delta >\frac{3}{2},}\hfill \end{array}\right.$$

where δ represents a real number.

For $\vartheta =2,$ we derive the subsequent corollary from Theorem 4.

Corollary 10.

Assume that $m\ge 1$ and $0\le \lambda <1.$ A function $h_m\in {\mathcal{S}}_{\Sigma }^{*,m}(\lambda ),$ then

$$|b_{m+1}| \le {\displaystyle \frac{\sqrt{2(1-\lambda )}}{m}},$$

$$|b_{2m+1}| \le {\displaystyle \frac{(m+1)(1-\lambda )}{m^2}}$$

and

$$|b_{2m+1}-\delta b_{m+1}| \le \left\{\begin{array}{cc}{\displaystyle \frac{2(1-\lambda )(m+1-2\delta )}{m^2}}\hfill & {\displaystyle \mathit{if}\delta <\frac{1}{2},}\hfill \\ {\displaystyle \frac{2(1-\lambda )}{2m}}\hfill & {\displaystyle \mathit{if}\frac{1}{2}\le \delta \le m+\frac{1}{2},}\hfill \\ {\displaystyle \frac{2(1-\lambda )(2\delta -1-m)}{2m^2}}\hfill & {\displaystyle \mathit{if}\delta >m+\frac{1}{2},}\hfill \end{array}\right.$$

where δ represents a real number.

For $m=1,$ we derive the subsequent corollary from Theorem 5.

Corollary 11.

Assume that $0\le \lambda <1$ and $\vartheta \ge 2m.$ A function $h\in {\mathcal{V}}_{\Sigma }(\vartheta ,\lambda ),$ and if $b_{n+1}=0,$$1\le n\le k-1,$ then

$$|b_{k+1}| \le {\displaystyle \frac{\vartheta (1-\lambda )}{k(k+1)}},\forall k\ge 2.$$

For the case where $m=1$ and $\vartheta =2,$ we derive the following corollary from Theorem 3.

Corollary 12.

Assume that $0\le \lambda <1.$ A function $h\in {\mathcal{C}}_{\Sigma }(\lambda ),$ and if $b_{n+1}=0,$$1\le n\le k-1,$ then

$$|b_{k+1}| \le {\displaystyle \frac{2(1-\lambda )}{k(k+1)}},\forall k\ge 2.$$

For $\vartheta =2,$ we derive the subsequent corollary from Theorem 3.

Corollary 13.

Assume that $m\ge 1$ and $0\le \lambda <1.$ A function $h_m\in {\mathcal{C}}_{\Sigma }^m(\lambda ),$ and if $b_{mn+1}=0,$$1\le n\le k-1,$ then

$$|b_{mk+1}| \le {\displaystyle \frac{2(1-\lambda )}{mk(mk+1)}},\forall k\ge 2.$$

For $m=1,$ we derive the subsequent corollary from Theorem 6.

Corollary 14.

Assume that $0\le \lambda <1$ and $\vartheta \ge 2.$ A function $h\in {\mathcal{V}}_{\Sigma }(\vartheta ,\lambda ),$ then

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

$$|b_3| \le {\displaystyle \frac{\vartheta (1-\lambda )}{2}}$$

and

$$|b_3-\delta b_2^2| \le \left\{\begin{array}{cc}{\displaystyle \frac{\vartheta (1-\lambda )(1-\delta )}{2}}\hfill & {\displaystyle \mathit{if}\delta <\frac{2}{3},}\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )}{6}}\hfill & {\displaystyle \mathit{if}\frac{2}{3}\le \delta \le \frac{4}{3},}\hfill \\ {\displaystyle \frac{\vartheta (1-\lambda )(\delta -1)}{2}}\hfill & {\displaystyle \mathit{if}\delta >\frac{4}{3},}\hfill \end{array}\right.$$

where δ represents a real number.

For $\vartheta =2,$ we derive the subsequent corollary from Theorem 6.

Corollary 15.

Assume that $m\ge 1$ and $0\le \lambda <1.$ A function $h_m\in {\mathcal{C}}_{\Sigma }^m(\lambda ),$ then

$$|b_{m+1}| \le \sqrt{{\displaystyle \frac{\vartheta (1-\lambda )}{m^2(m+1)}}},$$

$$|b_{2m+1}| \le {\displaystyle \frac{2(1-\lambda )}{2m^2}}$$

and

$$|b_{2m+1}-\delta b_{m+1}| \le \left\{\begin{array}{cc}{\displaystyle \frac{2(1-\lambda )(m+1-2\delta )}{2m^2(m+1)}}\hfill & {\displaystyle \mathit{if}\delta <\frac{{(m+1)}^2}{2(2m+1)},}\hfill \\ {\displaystyle \frac{2(1-\lambda )}{2(2m+1)}}\hfill & {\displaystyle \mathit{if}\frac{{(m+1)}^2}{2(2m+1)}\le \delta \le \frac{(m+1)(3m+1)}{2(2m+1)},}\hfill \\ {\displaystyle \frac{2(1-\lambda )(2\delta -1-m)}{2m^2(m+1)}}\hfill & {\displaystyle \mathit{if}\delta >\frac{(m+1)(3m+1)}{2(2m+1)},}\hfill \end{array}\right.$$

where δ represents a real number.

Remark 4. (i) Corollary 1, Corollary 6, and Corollary 11 confirm the coefficient bound established by Huo Tang [30].

(ii) Corollary 2, Corollary 7, and Corollary 12 confirm the coefficient bound established by Hamidi and Jahangiri [31].

(iii) Corollary 3 confirms the coefficient bound established by Sakar and Güney [32].

(iv) Corollary 8 confirms the coefficient bound established by Hamidi and Jahangiri [33].

(v) Corollary 10 and Corollary 15 confirm the bounds of $|b_2|$ and improve the bounds of $|b_3|$ established by Sivasubramanian [10].

(vi) Corollary 5 confirms the bounds of $|b_2|$ and $|b_3|$ established by Srivastava [11].

(vii) Corollary 4, Corollary 9, and Corollary 14 confirm the coefficient bound established by Sharma [27].

5. Conclusions

In the present work, we introduced three classes of m-fold symmetric bi-univalent functions with bounded boundary rotation. We applied the Faber polynomial technique and investigated the mth coefficient bounds, the initial coefficients, and the Fekete–Szegö for this newly defined class of m-fold symmetric functions.

Additionally, the research discussed in this article could be expanded by exploring specific types of functions and various operators applied to m-fold symmetric functions that exhibit bounded boundary rotation and bounded radius rotation. Additionally, similar results can be derived for other noteworthy special functions and various operators found in the literature.

The study considered in this article can be used in a lung CT image segmentation method that includes a new image enhancement model that uses the bounded turning Mittag–Leffler function with bounded boundary rotation to improve the CT images for much better segmentation outcomes [34]. The approximated coefficient constraints can be used in image processing, specifically texture analysis. This work also be extended for colored images and investigate various image-processing techniques like enhancement, sharpening, pattern identification, restoration, and retrieval. Mathematically, future research can be carried out with the results of Feketö inequality obtained for inverse functions and can be applied in image processing.