Coefficient estimates for a subclass of bi-univalent functions associated with the Salagean differential operator

In this paper, we present and examine a novel subset of the function class ∑, which consists of analytic and bi-univalent functions defined in the open unit disk U and connected to the Salagean differential operator. Additionally, we determine estimates for the Taylor-Maclaurin coefficients |a₂| and |a₃| functions within this new subclass and enhance some recent findings.

2010 Mathematics Subject Classification. 30C45; 30C50.

1. Introduction

Consider the class of functions A defined as

$f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}z+{\displaystyle \sum _{n\mathrm{<mo>=</mo>2}}^{\infty }}{a}_n{z}^n\mathrm{,}\text{ (1}\text{.1)}$

where f(z) is analytic in the open unit disk $\mathbb{U}\mathrm{<mo>=</mo>}z\in C\mathrm{<mo>:</mo><mo>|</mo>}z\mathrm{<mo>|</mo><mo><</mo>1}$ Let s be the subset of functions f∈A that are univalent in .

The Koebe one-quarter theorem [3] states that for every f∈s, image of under f contains a disk of radius $\frac{1}{4}$ . Thus, every f∈s has an inverse f^-1, defined as

$f^{-1}\mathrm{<mo\; stretchy="false">(</mo>}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>=</mo>}z\mathrm{,}z\in \mathbb{U}\mathrm{,}$

and

$f\mathrm{<mo\; stretchy="false">(</mo>}{f}^{-1}\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>=</mo>}w\mathrm{,}\text{ for }\mathrm{<mo>|</mo>}w\mathrm{<mo>|</mo><mo><</mo>}{r}_0\mathrm{<mo\; stretchy="false">(</mo>}f\mathrm{<mo\; stretchy="false">)</mo>,}\text{ where }{r}_0\mathrm{<mo\; stretchy="false">(</mo>}f\mathrm{<mo\; stretchy="false">)</mo>}\ge \frac{1}{4}\mathrm{,}$

with

$f^{-1}\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}g\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}w-a_2{w}^2+\mathrm{<mo\; stretchy="false">(</mo>2}{a}_2^2-a_3\mathrm{<mo\; stretchy="false">)</mo>}{w}^3-\mathrm{<mo\; stretchy="false">(</mo>5}{a}_2^3-5a_2{a}_3+a_4\mathrm{<mo\; stretchy="false">)</mo>}{w}^4+\dots \text{ (1}\text{.2)}$

A function f∈A is bi-univalent in $\mathbb{U}$ if both f and f^-1 are univalent in $\mathbb{U}$ . Let ∑ be the set of bi-univalent functions in given by [1,6].

Several authors have investigated bounds for various subclasses of biunivalent functions [2-5,7-14]. However, the estimation of the Taylor-Maclaurin coefficients |an| for $n\in \frac{\mathbb{N}}{\mathrm{<mo>\{</mo>1,2<mo>\}</mo>}}\mathrm{<mo>;</mo>}\mathbb{N}\mathrm{<mo>:</mo><mo>=</mo><mo>\{</mo>1,2,3,}\cdot \cdot \cdot \mathrm{<mo>\}</mo>}$ remains an open problem.

In 1983, Salagean [13] introduced the differential operator ${\mathcal{D}}^k\mathrm{<mo>:</mo>}\mathcal{A}\to \mathcal{A}$ defined by

${\mathcal{D}}^{0}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>,}$

${\mathcal{D}}^{1}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\mathcal{D}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}z{f}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>,}$

${\mathcal{D}}^{k}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\mathcal{D}\mathrm{<mo\; stretchy="false">(</mo>}{\mathcal{D}}^{k-1}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>=</mo>}z\mathrm{<mo\; stretchy="false">(</mo>}{\mathcal{D}}^{k-1}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}{)}^{\prime }\mathrm{,}k\in \mathbb{N}\mathrm{.}$

That is, the Salagean differential operator D^k applied to a function f(z) is defined recursively as the derivative of D^k-1f(z) multiplied by z. This operator is employed in the study of analytic functions, particularly for estimating coefficients of certain classes of functions. It should be noted that

${\mathcal{D}}^{k}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}z+{\displaystyle \sum _{n\mathrm{<mo>=</mo>2}}^{\infty }}{n}^k{a}_n{z}^n\mathrm{,}k\in {\mathbb{N}}_0\mathrm{<mo>=</mo><mo>\{</mo>0<mo>\}</mo>}\cup \mathbb{N}\mathrm{.}$

This paper aims to introduce a new subclass of the function class ∑ associated with the Salagean differential operator and derive estimates for the coefficients |a₂| and |a₃| functions within these new subclasses of the function class ∑.

2. The subclass ${\mathcal{S}}_{\Sigma }^{h\mathrm{,}p}$

In this section, we introduce and investigate the general subclass ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}$ .

Definition 2.1: Let the functions $h\mathrm{,}p\mathrm{<mo>:</mo>}\mathbb{U}\to ℂ$ be so constrained that

$\min \mathrm{<mo>\{</mo>}\Re \mathfrak{e}\mathrm{<mo\; stretchy="false">(</mo>}h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>,}\Re \mathfrak{e}\mathrm{<mo\; stretchy="false">(</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>\}</mo><mo>></mo>0,}z\in \mathbb{U}\mathrm{,}h\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>=</mo>1.}$

Also let the function f, defined by (1.1), be in the analytic function class A. we say that

$f\in {\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>,}k\in {\mathbb{N}}_0\mathrm{,0}\le \lambda \mathrm{<mo><</mo>1.}$

if the following conditions are satisfied:

$f\in \mathrm{:₃;}and\frac{{\mathcal{D}}^{k+1}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}{\mathcal{D}}^{k}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}+\lambda {\mathcal{D}}^{k+1}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}\in h\mathrm{<mo\; stretchy="false">(</mo>}\mathbb{U}\mathrm{<mo\; stretchy="false">)</mo>,}z\in \mathbb{U}\mathrm{,}\text{ (2}\text{.1)}$

and

$\frac{{\mathcal{D}}^{k+1}g\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}{\mathcal{D}}^{k}g\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo>}+\lambda {\mathcal{D}}^{k+1}g\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo>}}\in p\mathrm{<mo\; stretchy="false">(</mo>}\mathbb{U}\mathrm{<mo\; stretchy="false">)</mo>,}w\in \mathbb{U}\mathrm{.}\text{ (2}\text{.2)}$

where the function g(w) is given by (1.2).

Remark 2.2: There are many choices of the functions h(z) and p(z) which would provide interesting subclasses of the analytic function class A.

1. For  $h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}\frac{1+z}{1-z}{\mathrm{<mo\; stretchy="false">)</mo>}}^{\alpha }$ and k=l=0, we have  ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>0,0<mo\; stretchy="false">)</mo><mo>=</mo>}{\mathcal{S}}_{\mathrm{:₃;}}^{\ast }\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ and k=1,l=0,  ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>1,0<mo\; stretchy="false">)</mo><mo>=</mo>}{\mathcal{K}}_{\mathrm{:₃;}}\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ where the classes  ${\mathcal{S}}_{\mathrm{:₃;}}^{\ast }\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ and  ${\mathcal{K}}_{\mathrm{:₃;}}\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ of bi-starlike functions of order a and bi-convex functions of order a corresponding, was introduced and studied by Brannan and Taha [1].
2. For  $h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{1+\mathrm{<mo\; stretchy="false">(</mo>1}-2\beta \mathrm{<mo\; stretchy="false">)</mo>}z}{1-z}$ and k=l=0, we have  ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>0,0<mo\; stretchy="false">)</mo><mo>=</mo>}{\mathcal{S}}_{\mathrm{:₃;}}^{\ast }\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ and k=1,l=0,  ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>1,0<mo\; stretchy="false">)</mo><mo>=</mo>}{\mathcal{K}}_{\mathrm{:₃;}}\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ where the classes  ${\mathcal{S}}_{\mathrm{:₃;}}^{\ast }\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ and  ${\mathcal{K}}_{\mathrm{:₃;}}\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ of bi-starlike functions of order b and bi-convex functions of order b corresponding, was introduced and studied by Brannan and Taha [1].
3. For  $h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}\frac{1+z}{1-z}{\mathrm{<mo\; stretchy="false">)</mo>}}^{\alpha }$ we have  ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\mathcal{S}}_{\mathrm{:₃;}}^{k\mathrm{,}\lambda }\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ and taking $h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{1+\mathrm{<mo\; stretchy="false">(</mo>1}-2\beta \mathrm{<mo\; stretchy="false">)</mo>}z}{1-z}$ ,  ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}{\mathcal{S}}_{\mathrm{:₃;}}^{k\mathrm{,}\lambda }\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ where the classes  ${\mathcal{S}}_{\mathrm{:₃;}}^{k\mathrm{,}\lambda }\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ and  ${\mathcal{S}}_{\mathrm{:₃;}}^{k\mathrm{,}\lambda }\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ was introduced and studied by J.Jothibaso [6].

3. Coefficient estimates

For proof of the theorem, we need the following lemma.

Lemma 3.1: (see [3]). If p∈P, then |ck|≤2 for each k, where p is the family of all functions p(z) analytic in $\mathbb{U}$ for which $\Re \mathfrak{e}\mathrm{<mo\; stretchy="false">(</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo><mo>></mo>0,}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>1}+c_{1}z+c_2{z}^2+\cdot \cdot \cdot$ for $z\in \mathbb{U}$ .

Theorem 3.2: Let f(z) given by the Taylor Maclaurin series expansion (1.1) be in the class ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>,<mo\; stretchy="false">(</mo>0}\le \lambda \mathrm{<mo><</mo>1<mo\; stretchy="false">)</mo>}$ . Then,

$\mathrm{<mo>|</mo>}{a}_2\mathrm{<mo>|</mo>}\le \min \left\{\sqrt{\frac{\mathrm{<mo>|</mo>}{h}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2+\mathrm{<mo>|</mo>}{p}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2}{2^{2k+1}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}}\mathrm{,}\sqrt{\frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{2^{2k+2}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{8<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}}\right\}\mathrm{,}\text{ (3}\text{.1)}$

and

$\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le \min \mathrm{<mo>\{</mo>}\frac{\mathrm{<mo>|</mo>}{h}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2+\mathrm{<mo>|</mo>}{p}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2}{2^{2k+1}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{\mathrm{8<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{,}$

$\frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{\mathrm{8<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}+\frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{2^{2k+2}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{8<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{<mo>\}</mo>.}$

Proof. First of all, it follows from the conditions (2.1) and (2.2) that

$\frac{{\mathcal{D}}^{k+1}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}{\mathcal{D}}^{k}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}+\lambda {\mathcal{D}}^{k+1}f\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo>,}z\in \mathbb{U}\mathrm{,}\text{ (3}\text{.2)}$

and

$\frac{{\mathcal{D}}^{k+1}g\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo>}}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}{\mathcal{D}}^{k}g\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo>}+\lambda {\mathcal{D}}^{k+1}g\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo>,}w\in \mathbb{U}\mathrm{,}\text{ (3}\text{.3)}$

where the function g(w) is given by (1.2). respectively, where h(z) and p(w) satisfy the conditions of Definition (2.1). Furthermore, the functions h(z) and p(w) have the following Taylor-Maclaurin series expansions:

$h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>1}+h_{1}z+h_2{z}^2+\dots$

and

$p\mathrm{<mo\; stretchy="false">(</mo>}w\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>1}+p_{1}w+p_2{w}^2+\dots$

Now, equating the coefficients in (3.2) and (3.3), we get

$2^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}{a}_2\mathrm{<mo>=</mo>}{h}_1\mathrm{,}\text{ (3}\text{.4)}$

$2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}{a}_2^2+2\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}{a}_3\mathrm{<mo>=</mo>}{h}_2\mathrm{,}\text{ (3}\text{.5)}$

$-2^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}{a}_2\mathrm{<mo>=</mo>}{p}_1\mathrm{,}\text{ (3}\text{.6)}$

$2\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>2}{a}_2^2-a_3\mathrm{<mo\; stretchy="false">)</mo>}+2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}{a}_2^2\mathrm{<mo>=</mo>}{p}_2\mathrm{.}\text{ (3}\text{.7)}$

From (3.4) and (3.6), we obtain

$h_1\mathrm{<mo>=</mo>}-p_1\mathrm{,}$

and

$h_1^2+p_1^2{\mathrm{<mo>=</mo>2}}^{2k+1}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2{a}_2^2\mathrm{.}\text{ (3}\text{.8)}$

Also, From (3.5) and (3.7), we find that

$h_2+p_2{\mathrm{<mo>=</mo>2<mo\; stretchy="false">(</mo>2}}^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+2\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">)</mo>}{a}_2^2\mathrm{.}\text{ (3}\text{.9)}$

Therefore, we find from the equations (3.8) and (3.9) that

$\mathrm{<mo>|</mo>}{a}_2{\mathrm{<mo>|</mo>}}^2\le \frac{\mathrm{<mo>|</mo>}{h}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2+\mathrm{<mo>|</mo>}{p}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2}{2^{2k+1}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}\mathrm{,}$

and

$\mathrm{<mo>|</mo>}{a}_2{\mathrm{<mo>|</mo>}}^2\le \frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{2^{2k+2}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+2^3\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}$

respectively. So we get the desired estimate on the coefficient |a₂| as asserted in (3.1). Next, to find the bound on the coefficient |a₃|, we subtract (3.7) from (3.5). We thus get

$h_2-p_2\mathrm{<mo>=</mo>4}\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>}{a}_3-a_2^2\mathrm{<mo\; stretchy="false">)</mo>.}\text{ (3}\text{.10)}$

Upon substituting the value of $a_2^2$ from (3.8) into (3.10), it follows that

$a_3\mathrm{<mo>=</mo>}\frac{h_1^2+p_1^2}{2^{2k+1}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{h_2-p_2}{4\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.}$

We thus find that

$\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le \frac{\mathrm{<mo>|</mo>}{h}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2+\mathrm{<mo>|</mo>}{p}^{\prime }{\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}^2}{2^{2k+1}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{8\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.}$

On the other hand, upon substituting the value of $a_2^2$ from (3.9) into (3.10), it follows that

$a_3\mathrm{<mo>=</mo>}\frac{h_2-p_2}{4\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}+\frac{h_2+p_2}{4\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}+2^{2k+1}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}}\mathrm{.}$

Consequently, we have

$\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le \frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{8\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}+\frac{\mathrm{<mo>|</mo>}{h^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}+\mathrm{<mo>|</mo>}{p^{\prime }}^{\prime }\mathrm{<mo\; stretchy="false">(</mo>0<mo\; stretchy="false">)</mo><mo>|</mo>}}{8\cdot 3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}+2^{2k+2}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}}\mathrm{.}$

This evidently completes the proof of Theorem 3.2.

4. Corollaries and consequences

By setting $h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}\frac{1+z}{1-z}{\mathrm{<mo\; stretchy="false">)</mo>}}^{\alpha }$ , (0<α≤1) in Theorem 3.2. we get the following consequence.

Corollary 4.1: Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>}$ , (0<1). Then

$\mathrm{<mo>|</mo>}{a}_2\mathrm{<mo>|</mo>}\le \{\begin{array}{cc}\frac{2\alpha }{\sqrt{2^{2k+1}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{4<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}}\mathrm{,}& k\mathrm{<mo>=</mo>0,}\\ \frac{2\alpha }{2^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{,}& k\mathrm{<mo>=</mo>1,2,3,}\dots \\ & \end{array}$

and

$\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le \{\begin{array}{cc}\frac{2{\alpha }^2}{2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>2}-2\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}+\frac{{\alpha }^2}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{,}& k\mathrm{<mo>=</mo>0,}\\ \frac{4{\alpha }^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{{\alpha }^2}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{,}& k\mathrm{<mo>=</mo>1,2,3,}\dots \\ & \end{array}$

Remark 4.2: Corollary 4.1 is an improvement of the following estimates obtained by J.Jothibaso [6].

Corollary 4.3: (see[6]) Let the function f(z) given by the Taylor-Maclaurin series expansion (1) be in the bi-univalent function class ${\mathcal{S}}_{\mathrm{:₃;}}^{k\mathrm{,}\lambda }\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ . Then

$\mathrm{<mo>|</mo>}{a}_2\mathrm{<mo>|</mo>}\le \frac{2\alpha }{\sqrt{4\alpha \mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k+\mathrm{<mo\; stretchy="false">[</mo>2}\alpha \mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}-\mathrm{<mo\; stretchy="false">(</mo>}\alpha -\mathrm{1<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>}}^2{\mathrm{<mo\; stretchy="false">]</mo>2}}^{2k}}}\mathrm{,}$

and

$\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le \frac{4{\alpha }^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{\alpha }{3^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{.}$

Remark 4.4: It is easy to see that [(i)]

1. For the coefficient |a₂|, If k = 0 and 0<α≤1, we have

$\frac{2\alpha }{\sqrt{2^{2k+1}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{4<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}}\le \frac{2\alpha }{\sqrt{4\alpha \mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k+\mathrm{<mo\; stretchy="false">[</mo>2}\alpha \mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}-\mathrm{<mo\; stretchy="false">(</mo>}\alpha -\mathrm{1<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>}}^2{\mathrm{<mo\; stretchy="false">]</mo>2}}^{2k}}}\mathrm{.}$

In another case, if k = 1,2,3… and 0<α≤1, we have

$\frac{2\alpha }{2^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\le \frac{2\alpha }{\sqrt{4\alpha \mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k+\mathrm{<mo\; stretchy="false">[</mo>2}\alpha \mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}-\mathrm{<mo\; stretchy="false">(</mo>}\alpha -\mathrm{1<mo\; stretchy="false">)</mo><mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>}}^2{\mathrm{<mo\; stretchy="false">]</mo>2}}^{2k}}}\mathrm{.}$

2. For the coefficient |a₃|, we make the following observations: If k=0 and 0<α≤1, we have

$\frac{2{\alpha }^2}{2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>2}-2\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\le \frac{4{\alpha }^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}\mathrm{,}$

and

$\frac{{\alpha }^2}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\le \frac{\alpha }{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{.}$

Then

$\frac{2{\alpha }^2}{2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>2}-2\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}+\frac{{\alpha }^2}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\le \frac{4{\alpha }^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{\alpha }{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{.}$

In another case, if k = 1,2,3… and 0<1, we have

$\frac{{\alpha }^2}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\le \frac{\alpha }{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{.}$

Then

$\frac{4{\alpha }^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{{\alpha }^2}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\le \frac{4{\alpha }^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{\alpha }{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{.}$

Thus Theorem 3.2 clearly improves the estimate of coefficients |a₂| and |a₃| obtained by J.Jothibaso [6].

By setting $h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo><mo\; stretchy="false">(</mo>}\frac{1+z}{1-z}{\mathrm{<mo\; stretchy="false">)</mo>}}^{\alpha }$ , k=λ=0 in Theorem 3.2. we get the following consequence.

Corollary 4.5: Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class ${\mathcal{S}}_{\mathrm{:₃;}}^{\ast }\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ . Then

$\mathrm{<mo>|</mo>}{a}_2\mathrm{<mo>|</mo>}\le \sqrt{2}\alpha \mathrm{,}and\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le 3{\alpha }^2\mathrm{.}$

Remark 4.6: Corollary 4.5 is an improvement of the following estimates obtained by the coefficient estimates for a well-known class ${\mathcal{S}}_{\mathrm{:₃;}}^{\ast }\mathrm{<mo\; stretchy="false">(</mo>}\alpha \mathrm{<mo\; stretchy="false">)</mo>}$ of strongly bi-starlike functions of order α as in [1].

By setting $h\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}p\mathrm{<mo\; stretchy="false">(</mo>}z\mathrm{<mo\; stretchy="false">)</mo><mo>=</mo>}\frac{1+\mathrm{<mo\; stretchy="false">(</mo>1}-2\beta \mathrm{<mo\; stretchy="false">)</mo>}z}{1-z}\mathrm{,<mo\; stretchy="false">(</mo>0}\le \beta \mathrm{<mo><</mo>1,<mi>\; </mi>}z\in \mathbb{U}\mathrm{<mo\; stretchy="false">)</mo>}$ in Theorem 3.2. we get the following consequence.

Corollary 4.7: Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class ${\mathcal{S}}_{\mathrm{:₃;}}^{h\mathrm{,}p}\mathrm{<mo\; stretchy="false">(</mo>}k\mathrm{,}\lambda \mathrm{<mo\; stretchy="false">)</mo>}$ , (0≤<1). Then

$\mathrm{<mo>|</mo>}{a}_2\mathrm{<mo>|</mo>}\le \min \left\{\frac{\mathrm{2<mo\; stretchy="false">(</mo>1}-\beta \mathrm{<mo\; stretchy="false">)</mo>}}{2^k\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}\mathrm{<mi>\; </mi><mi>\; </mi>,<mi>\; </mi><mi>\; </mi>}\sqrt{\frac{\mathrm{2<mo\; stretchy="false">(</mo>1}-\beta \mathrm{<mo\; stretchy="false">)</mo>}}{2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>2}-2\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}}\right\}\mathrm{,}$

and

$\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le \min \left\{\frac{\mathrm{4<mo\; stretchy="false">(</mo>1}-\beta {\mathrm{<mo\; stretchy="false">)</mo>}}^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{1-\beta }{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{<mi>\; </mi>,<mi>\; </mi>}\frac{\mathrm{2<mo\; stretchy="false">(</mo>1}-\beta \mathrm{<mo\; stretchy="false">)</mo>}}{2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>2}-2\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}+\frac{1-\beta }{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\right\}\mathrm{.}$

Remark 4.8: Corollary 4.7 is an improvement of the following estimates obtained by J.Jothibaso [6].

Corollary 4.9: (see[6]) Let the function f(z) given by the Taylor-Maclaurin series expansion (1.1) be in the bi-univalent function class ${ℳ}_{\mathrm{:₃;}}^{k\mathrm{,}\lambda }\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ . Then

$\mathrm{<mo>|</mo>}{a}_2\mathrm{<mo>|</mo>}\le \sqrt{\frac{\mathrm{2<mo\; stretchy="false">(</mo>1}-\beta \mathrm{<mo\; stretchy="false">)</mo>}}{2^{2k}\mathrm{<mo\; stretchy="false">(</mo>}{\lambda }^2-\mathrm{1<mo\; stretchy="false">)</mo>}+\mathrm{<mo\; stretchy="false">(</mo>2}-2\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}}\mathrm{,}$

and

$\mathrm{<mo>|</mo>}{a}_3\mathrm{<mo>|</mo>}\le \frac{\mathrm{4<mo\; stretchy="false">(</mo>1}-\beta {\mathrm{<mo\; stretchy="false">)</mo>}}^2}{2^{2k}{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda \mathrm{<mo\; stretchy="false">)</mo>}}^2}+\frac{1-\beta }{\mathrm{<mo\; stretchy="false">(</mo>1}-\lambda {\mathrm{<mo\; stretchy="false">)</mo>3}}^k}\mathrm{.}$

Corollary 4.10: By setting k=λ=0 in Corollary 4.7, we have the coefficients estimates for the well-known class ${\mathcal{S}}_{\mathrm{:₃;}}^{\ast }\mathrm{<mo\; stretchy="false">(</mo>}\beta \mathrm{<mo\; stretchy="false">)</mo>}$ of bi-starlike functions of order  as in [1]. Further, taking k=1,λ=0 in Corollary 4.7, we obtain the estimates for the well-known class k_∑(β) of bi-convex functions of order  and our results reduce to. [1].

Conclusion

This paper introduces a new subclass of the function class ∑ involving analytic and bi-univalent functions associated with the Salagean differential operator. Our study provides estimates for the Taylor-Maclaurin coefficients |a₂| and |a₃| for functions within this subclass, contributing to the advancement of knowledge in this area. The findings enhance recent research in the field and open up new avenues for further exploration and development in the theory of analytic and bi-univalent functions.

A part of this research was carried out while the Third author was visiting the University of Alberta.

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