On Third Hankel Determinant for Certain Subclass of Bi-Univalent Functions

: This study presents a subclass 𝒮(𝛽) of bi-univalent functions within the open unit disk region 𝐷 . The objective of this class is to determine the bounds of the Hankel determinant of order 3, ( Ⱨ (cid:2871) (1) ) . In this study, new constraints for the estimates of the third Hankel determinant for the class 𝒮(𝛽) are presented, which are of considerable interest in various ﬁ elds of mathematics, including complex analysis and geometric function theory. Here, we de ﬁ ne these bi-univalent functions as 𝒮(𝛽) and impose constraints on the coe ﬃ cients │𝑎 (cid:3041) │ . Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, speci ﬁ cally for n = 2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These ﬁ ndings contribute to the broader understanding of bi-univalent functions and their potential applications in diverse mathematical contexts. Notably, the results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses.


Introduction
Let  indicate the collection of functions  analytic in the open unit disk  = ᶎ: ᶎ ∈ ℂ and |ᶎ| < 1 .An analytic function  ∈  has Taylor series expansion of the form: The class of all functions in  which are univalent in  is denoted by .The Koebe-One-Quarter Theorem [1] ensures that the image of  under each  ∈  contains a disk of radius A function  ∈ ∑ is said to be bi-univalent in  if both (ᶎ) and  (ᶎ) are univalent in .
In 1967, Lewin [2] obtained a coefficient bound that is given by | | < 1.51 for all function  ∈ ∑ of the form (1), and he looked at the class ∑ of bi-univalent functions in .In 1967, Clunie and Brannan [3] conjectured that | | ≤ √2 for  ∈ ∑.After that, mates on the first two Taylor-Maclaurin coefficients were found in these subclasses (see [7][8][9][10]).Several authors introduced initial Maclaurin coefficients bounds for subclasses of bi-univalent functions (see [11,12]).Many researchers ([11,13,14]) have studied numerous curious subclasses of the bi-univalent function class Ω and observed non-sharp bounds on the first two Taylor-Maclaurin coefficients.As well as this, the coefficient problem for all of the Taylor-Maclaurin coefficients | |, n = 3,4,... is as yet an open problem ( [2]).Also, let  represent the class of analytic functions  that are normalized by the condition: Noonan and Thomas [15]  , (  = ).
By applying triangle inequality for Ⱨ  (), we have Our paper provides a subclass () of bi-univalent functions within the open unit disk region .The objective of this class is to determine the bounds of the Hankel determinant of order 3, (Ⱨ (1)).In this study, new constraints for the estimates of the third Hankel determinant for the class () are presented.
The subsequent lemmas are important for establishing our results:

Main Results
Definition 1.A function  belonging to the class ∑, as defined by equation ( 1) is considered to be in the class () if it fulfills the following requirement: and where (0 <  ≤ 1), ᶎ,  ∈  and  =  .

Conclusions
This article presented a comprehensive investigation of the third Hankel determinant H3(1) for a certain subclass of bi-univalent functions, ().This subclass is of significant interest in various mathematical fields, including complex analysis and geometric function theory.We defined the bi-univalent functions () and imposed constraints on the coefficients │ │.Our findings provided the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n = 2, 3, 4, and 5. Furthermore, we advanced the understanding of these functions by deriving the third Hankel determinant for this particular class, which revealed several intriguing scenarios.This achievement led to the improvement of the bound of the third Hankel determinant for the class of bi-univalent functions ().Our study contributes to the broader understanding of bi-univalent functions, their subclasses, and their potential applications in diverse mathematical contexts.The results obtained may serve as a foundation for future investigations into the properties and applications of bi-univalent functions and their subclasses.Future research endeavors could explore further refinements of the bounds, as well as examine other subclasses of bi-univalent functions to uncover novel insights into their characteristics and potential applications.Ultimately, this study paves the way for a deeper exploration of the fascinating world of bi-univalent functions and their role in the realm of mathematics.
defined the  Hankel determinant of , in 1976 for  ≥ 1 and  ≥ 1 by