Coefficient Bounds for a Certain Subclass of Bi-Univalent Functions Associated with Lucas-Balancing Polynomials

Abstract

In this paper, we introduce a new subclass of bi-univalent functions defined using Lucas-Balancing polynomials. For functions in each of these bi-univalent function subclasses, we derive estimates for the Taylor–Maclaurin coefficients $\left|a_2\right|$ and $\left|a_3\right|$ and address the Fekete–Szegö functional problems for functions belonging to this new subclass. We demonstrate that several new results can be derived by specializing the parameters in our main findings. The results obtained from this study will enrich the theoretical foundation of this field and open new avenues for mathematical inquiry and application.

1. Introduction

The concept of balancing numbers $\left(B_n\right),n\ge 0$ was originally introduced by Behera and Panda [1]. These numbers are defined by the recurrence relation $B_{n+1}=6B_n-B_{n-1}$ for $n\ge 1$, with initial values set at $B_0=0$ and $B_1=1$. A related sequence, the Lucas-Balancing numbers, denoted as $C_n=\sqrt{8B_n^2+1}$, has garnered significant attention. Similar to $B_n$, they also satisfy the recurrence relation $C_{n+1}=6C_n-C_{n-1}$ for $n\ge 1$, and have initial terms $C_0=1$ and $C_1=3$. These numbers have since been subject to diverse generalizations and explored through various approaches, including investigations into sum and ratio formulas for balancing numbers, hybrid convolutions, the representation of sums using binomial coefficients, various methods for summing balancing numbers, reciprocals of sequences related to these numbers, incomplete balancing and Lucas-Balancing numbers, generating functions, and matrix-based methods for studying these sequences. Some references extend the concept to generalized balancing sequences, offering diverse insights and approaches to this mathematical topic. For more details, we refer readers to [2,3,4,5,6,7,8,9,10,11].

A natural progression in this line of inquiry involves the study of sequences of Lucas-Balancing polynomials denoted as $\left(C_n\left(x\right)\right)n\ge 0$ where $x\in \mathbb{C}$ was introduced in [7]. These polynomials can be recursively defined as follows

$$\begin{array}{cc}\hfill & C_0\left(x\right)=1,\hfill \\ \hfill & C_1\left(x\right)=3x,\hfill \\ \hfill & C_2\left(x\right)=18x^2-1,\mathrm{and}\hfill \\ \hfill & C_n\left(x\right)=6x{C}_{n-1}\left(x\right)-C_{n-2}\left(x\right),n\ge 2.\hfill \end{array}$$

(1)

In [12], the generating function of the Lucas-Balancing polynomials is denoted as $\mathcal{B}(x,z)$ and given by

$$\mathcal{B}(x,z)=\sum _{n=0}^{\infty }{C}_n\left(x\right)z^n=\frac{1-3xz}{1-6xz+z^2},$$

(2)

where $x\in [-1,1]$ and z is in the open unit disk $\mathbb{D}=\{z:z\in \mathbb{C}\mathrm{and}\left|z\right|<1\}$.

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

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

(3)

Furthermore, a single-valued analytic function f in a simply connected domain $\mathbb{D}$ is said to be univalent (Schlicht or simple) if it is injective. Let $\mathcal{S}$ represent the set of all functions $f\in \mathcal{A}$, that are univalent in $\mathbb{D}$. A function f is subordinated to g, denoted as $f\prec g$ if there exists an analytic Schwarz function $h\left(z\right)$ defined in $\mathbb{D}$, such that $f\left(z\right)=g\left(h\right(z\left)\right)$ with $h\left(0\right)=0$, and $\left|h\right(z\left)\right|\le 1$. In particular, if the function g is univalent in $\mathbb{D}$, then the following equivalence is valid [13]

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

and

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

According to the Koebe one-quarter theorem [14], for any function $f\in \mathcal{S}$, the image of $f\left(\mathbb{D}\right)$ contains the disk that is centered at 0 and of radius $\frac{1}{4}$. Thus, each function $f\in \mathcal{S}$ retains an inverse $f^{-1}:f\left(\mathbb{D}\right)\to \mathbb{D}$, for which the following conditions are met

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

and

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

In fact, $f^{-1}$ has the series expansion of the form

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

(4)

A function $f\in \mathcal{A}$ is said to be bi-univalent in $\mathbb{D}$, if both the function f and its inverse $f^{-1}$, are one-to-one in $\mathbb{D}$. Let $\Sigma$ represent the set of bi-univalent functions in $\mathbb{D}$ as defined by Equation (3). The following functions

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

with their respective inverses

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

are bi-univalent. However, the Koebe function, denoted as $K\left(z\right)=\frac{z}{{(1-z)}^2}$, does not belong to the class $\Sigma$, because it maps the open unit disk $\mathbb{D}\subset \mathbb{C}$ to $\mathrm{K}\left(\mathbb{D}\right)=\mathbb{C}\setminus (-\infty ,-\frac{1}{4}]$, which does not include $\mathbb{D}$. Other commonly encountered univalent functions that do not belong to $\Sigma$ include

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

For $0\le ϵ<1$, the most significant and thoroughly investigated subclasses of $\mathcal{S}$ are the class ${\mathcal{S}}^{*}\left(ϵ\right)$ of starlike functions of order $ϵ$ and the class, $\mathcal{K}\left(ϵ\right)$ of convex functions of order $ϵ$ in $\mathbb{D}$, which are defined by

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

In 1933, Fekete and Szegö introduced an inequality for the coefficients of univalent analytic functions, represented by the generalized functional $F^{\eta }\left(f\right)=a_{{}_3}-\eta a_{{}_2}^2$ where $0\le \eta <1$ [15]. This result, known as the Fekete–Szegö inequality, states that for $f\in \Sigma$ is given by (3), $|F^{\eta }\left(f\right)|\le 1+2e^{\frac{-2\eta }{1-\eta }}$ with $|F^{\eta }\left(f\right)|\le 1$ as $\eta$ approaches $-1$.

Recently, a multitude of authors have made significant strides in establishing tight coefficient bounds for diverse subclasses of bi-univalent functions, often intertwined with specific polynomial families, (see [16,17,18,19,20,21,22,23,24,25,26,27,28,29,30]).

2. Coefficient Bounds of the Class ${\mathcal{N}}_{\Sigma }^{\mathit{\lambda }}\left(\mathcal{B}(\mathit{x},\mathit{z})\right)$

We would like to introduce a new subclass of bi-univalent functions, which will be referred to as follows.

Definition 1.

Let $\lambda \in [0,1]$ and $x\in (\frac{1}{2},1]$. A function $f\in \Sigma$ given by (3) is said to be in the class ${\mathcal{N}}_{\Sigma }^{\lambda }\left(\mathcal{B}(x,z)\right)$ if the following subordinations are satisfied

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

(5)

and

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

(6)

where the function $g\left(w\right)=f^{-1}\left(w\right)$ is defined by (4) and $\mathcal{B}(x,z)$ is the generating function of the Lucas-Balancing polynomials given by (2).

Example 1.

A bi-univalent function $f\in \Sigma$ is said to be in the class ${\mathcal{N}}_{\Sigma }^0\left(\mathcal{B}(x,z)\right)={\mathcal{S}}_{\Sigma }^{*}(x,z)$, if the following subordination conditions hold:

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

(7)

and

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

(8)

where the function $g=f^{-1}$ is defined by (4).

Example 2.

A bi-univalent function $f\in \Sigma$ is said to be in the class ${\mathcal{N}}_{\Sigma }^1\left(\mathcal{B}(x,z)\right)={\mathcal{K}}_{\Sigma }(x,z)$, if the following subordination conditions hold:

$$1+\frac{z{f}^{\prime \prime }\left(z\right)}{f^{\prime }\left(z\right)}\prec \mathcal{B}(x,z)$$

(9)

and

$$1+\frac{w{g}^{\prime \prime }\left(w\right)}{g^{\prime }\left(w\right)}\prec \mathcal{B}(x,w),$$

(10)

where the function $g=f^{-1}$ is defined by (4).

Let $\Omega$ be the class of all analytic functions $\omega \in \mathbb{D}$ which satisfies that $\omega \left(0\right)=0$ and $\left|\omega \right(z\left)\right|<1$ for all $z\in \mathbb{D}$. We begin by recalling the following lemma, which will be instrumental in establishing the main result. Subsequently, we will present the coefficient estimates for the class ${\mathcal{N}}_{\Sigma }^{\lambda }\left(\mathcal{B}(x,z)\right)$ given in Definition 1.

Lemma 1

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

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

Theorem 1.

Let $f\in \Sigma$ of the form (3) be in the class ${\mathcal{N}}_{\Sigma }^{\lambda }\left(\mathcal{B}(x,z)\right)$Ṫhen

$$\left|a_2\right|\le \frac{3x\sqrt{3x}}{\sqrt{(1+\lambda )\left|9x^2-(1+\lambda )(18x^2-1)\right|}},$$

(11)

and

$$\left|a_3\right|\le \frac{27x^3}{(1+\lambda )\left|9x^2-(1+\lambda )(18x^2-1)\right|}+\frac{3x}{2(1+2\lambda )}.$$

(12)

Proof. 

Let $f\in {\mathcal{N}}_{\Sigma }^{\lambda }\left(\mathcal{B}(x,z)\right)$ for some $0\le \lambda \le 1$, and from (5) and (6), we have

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

(13)

and

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

(14)

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

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

Using Lemma 1, we obtain

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

(15)

Substituting $\mathcal{B}(x,z)$ defined in (2), on the right-hand sides of Equations (13) and (14), we obtain

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

(16)

and

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

(17)

Hence, Equations (13) and (14), become

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

(18)

and

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

(19)

By setting the coefficients in Equations (18) and (19) equal to each other, we obtain

$$(1+\lambda )a_2=C_1\left(x\right)c_1,$$

(20)

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

(21)

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

(22)

and

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

(23)

Using (20) and (22), we obtain the following equations

$$c_1=-d_1$$

(24)

and

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

(25)

Furthermore, employing Equations (21), (23) and (25) yields

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

(26)

By utilizing Lemma 1, and examining Equations (20) and (24), we can be deduce

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

(27)

therefore

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

(28)

Substituting $C_1\left(x\right)$ and $C_2\left(x\right)$ provided in (1) into Equation (28) leads to the following

$$\left|a_2\right|\le \frac{3x\sqrt{3x}}{\sqrt{(1+\lambda )\left|9x^2-(1+\lambda )(18x^2-1)\right|}}.$$

Subtracting Equation (23) from Equation (21) we have

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

(29)

This leads to the following inequality

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

(30)

Utilizing Lemma 1 and employing Equation (1) we obtain

$$\left|a_3\right|\le \frac{27x^3}{(1+\lambda )\left|9x^2-(1+\lambda )(18x^2-1)\right|}+\frac{3x}{2(1+2\lambda )}.$$

(31)

This completes the proof of Theorem 1. □

3. Fekete–Szegö Functional Estimations of the Class ${\mathcal{N}}_{\Sigma }^{\mathit{\lambda }}\left(\mathcal{B}(\mathit{x},\mathit{z})\right)$

In this section, employing the values of $a_2^2$ and $a_3$ aids in deriving the Fekete–Szegö inequality for functions $f\in {\mathcal{N}}_{\Sigma }^{\lambda }\left(\mathcal{B}(x,z)\right)$.

Theorem 2.

Let $f\in \Sigma$ given by the form (3) be in the class ${\mathcal{N}}_{\Sigma }^{\lambda }\left(\mathcal{B}(x,z)\right)$. Then

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}\frac{3x}{2(1+2\lambda )}\hfill & if0\le \left|h\left(\eta \right)\right|\le \frac{1}{4(1+2\lambda )}\hfill \\ 6x\left|h\left(\eta \right)\right|\hfill & if\left|h\left(\eta \right)\right|\ge \frac{1}{4(1+2\lambda )},\hfill \end{array}\right.$$

where

$$h\left(\eta \right)=\frac{9x^2(1-\eta )}{2(1+\lambda )\left[9x^2-(1+\lambda )\left(18x^2-1\right)\right]}.$$

Proof. 

From Equations (26) and (29), we have

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

where

$$h\left(\eta \right)=\frac{{\left(C_1\left(x\right)\right)}^2(1-\eta )}{2(1+\lambda )\left[{\left(C_1\left(x\right)\right)}^2-(1+\lambda )\left(C_2\left(x\right)\right)\right]}.$$

Then, considering (1) and applying (15), we can deduce that

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}\frac{3x}{2(1+2\lambda )}\hfill & \mathrm{if}0\le \left|h\left(\eta \right)\right|\le \frac{1}{4(1+2\lambda )}\hfill \\ 6x\left|h\left(\eta \right)\right|\hfill & \mathrm{if}\left|h\left(\eta \right)\right|\ge \frac{1}{4(1+2\lambda )}.\hfill \end{array}\right.$$

This completes the proof of Theorem 2. □

Corollary 1.

Let $f\in \Sigma$ given by the form (3) be in the class ${\mathcal{S}}_{\Sigma }^{*}(x,z)$. Then

$$\left|a_2\right|\le \frac{3x\sqrt{3x}}{\sqrt{\left|1-9x^2\right|}},$$

$$\left|a_3\right|\le \frac{27x^3}{\left|1-9x^2\right|}+\frac{3x}{2},$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}\frac{3x}{2}\hfill & if0\le \left|h_1\left(\eta \right)\right|\le \frac{1}{4}\hfill \\ 6x\left|h_1\left(\eta \right)\right|\hfill & if\left|h_1\left(\eta \right)\right|\ge \frac{1}{4},\hfill \end{array}\right.$$

where

$$h_1\left(\eta \right)=\frac{9x^2(1-\eta )}{2\left(1-9x^2\right)}.$$

Corollary 2.

Let $f\in \Sigma$ given by the form (3) be in the class ${\mathcal{K}}_{\Sigma }(x,z)$. Then

$$\left|a_2\right|\le \frac{3x\sqrt{3x}}{\sqrt{2\left|2-27x^2\right|}},$$

$$\left|a_3\right|\le \frac{27x^3}{2\left|2-27x^2\right|}+\frac{x}{2},$$

and

$$\left|a_3-\eta a_2^2\right|\le \left\{\begin{array}{cc}\frac{x}{2}\hfill & if0\le \left|h_2\left(\eta \right)\right|\le \frac{1}{12}\hfill \\ 6x\left|h_2\left(\eta \right)\right|\hfill & if\left|h_2\left(\eta \right)\right|\ge \frac{1}{12},\hfill \end{array}\right.$$

where

$$h_2\left(\eta \right)=\frac{9x^2(1-\eta )}{4\left(2-27x^2\right)}.$$

4. Conclusions

In our present investigation, we have introduced and studied the coefficient problems associated with the newly introduced subclasses. We have derived estimates of the Taylor–Maclaurin coefficients $|a_2|$, $|a_3|$ and Fekete–Szegö functional problems for functions belonging to these subclasses.