Certain New Subclassess of Analytic and Bi-univalent Functions

. The paper presents two novel subclasses of the function class Σ , which consists of bi-univalent functions deﬁned in the open unit disk D = { ζ : | ζ | < 1 } . The authors investigate the properties of these new subclasses and provide estimates for the absolute values of the second, third, and fourth Taylor-Maclaurin coeﬃcients r 2 , r 3 , and r 4 for functions in these subclasses.


Introduction
Let A be the class of functions with the following form: which are analytic in the open unit disc D = {ζ : |ζ| < 1}.
Lewin's investigation of the bi-univalent function class Σ led to the result that the absolute value of the second coefficient, denoted as r 2 , satisfies the inequality |r 2 | < 1.51.
In 1969, Netanyahu [6] showed that the maximum value of |r 2 | among all functions in Σ is 4 3 . In summary, Lewin showed that |r 2 | < 1.51 for functions in the class Σ, Netanyahu demonstrated that the maximum value of |r 2 | in Σ is 4 3 , and Brannan and Taha introduced subclasses of Σ akin to the starlike and convex function subclasses.
The motivation for the current paper is derived from the groundbreaking work conducted by Brannan and Taha [1], which has reignited interest in the exploration of analytic and bi-univalent functions in the field of mathematics in recent years. The success of Brannan and Taha work has served as inspiration and has generated an extensive number of subsequent research papers by other authors ( [2], [3], [5], [9], [11], [12]).
The objective of the present paper is to introduce two novel subclasses within the function class Σ and provide estimates for the coefficients |r 2 |, |r 3 | and |r 4 | for functions belonging to these newly defined subclasses. To achieve our main results, we need to revisit and state the following lemma. and satisfy h(0) = 1 and R(h(ζ)) > 0 for all ζ ∈ D. If a function h ∈ H is given by h(ζ) = 1 + d 1 ζ + d 2 ζ 2 + · · · for ζ ∈ D, then |d k | ≤ 2 for all k ∈ N.

4)
, and |r 4 | ≤ 4µ 3 + 2µ (2.6) Proof:We can express the inequalities stated in equations (2.1) and (2.2) in a simpler form as respectively, where m(ζ) and n(η) satisfy the following inequalities: Furthermore, the functions m(ζ) and n(η) have the forms By comparing the coefficients in equations (2.7) and (2.8), we obtain the following result.
which implies Adding (2.12) and (2.15), we obtain (2.21) By utilizing Lemma 1.1 for the coefficients m 2 and n 2 , we can directly derive the following This gives the bound on |r 2 | as asserted in (2.4).
To determine the bound for r 3 , we can subtract equation (2.15) from equation (2.12), we get By substituting the value of r 2 2 from equation (2.19) and noticing that m 2 1 = n 2 1 , we get the following (2.24) By utilizing Lemma 1.1 for the coefficients m 1 , m 2 , n 1 and n 2 , we readily get To determine the bound on |r 4 |, subtracting (2.16) from (2.13) with m 1 = −n 1 gives substitute the values of r 2 and r 3 from (2.17) and (2.24) in (2.25).
By utilizing Lemma 1.1 for the coefficients m 1 , m 2 and n 2 , we get By setting b = c = 1 in Theorem 2.1, we obtain the following corollary. Putting γ = 0, in Corollary 2.1, we get the following corollary. and where the function ψ is given by (2.3).
By selecting specific parameter values, researchers have identified several significant subclasses that have been extensively studied in previous papers. Here are some examples of these subclasses.
Next, we find the estimates on the cofficients |r 2 |, |r 3 | and |r 4 | for function in the class B Σ (γ, b, c, ρ).
By utilizing Lemma 1.1 for the coefficients m 1 , m 2 , n 1 and n 2 , we get (3.20) To determine the bound on |r 4 |, subtracting (3.13) from (3.10) and substitute the values of r 2 and r 3 from (3.14) and (3.19).
After Applying Lemma 1.1 for the cofficients m 1 , m 2 and n 2 , we get By setting b = c = 1 in Theorem 3.1, we have the following corollary. Putting γ = 0, in Corollary 3.1, we get the following corollary.

Conclusions
The primary focus of this paper is to introduce novel subclasses of bi-univalent functions in open unit disc D. Furthermore, we present upper limits for the coefficients |r 2 | , |r 3 | and |r 4 | for functions belonging to these new subclasses and their respective subclasses.

Conflicts of Interest:
The authors declare that there are no conflicts of interest regarding the publication of this paper.