On the Third Hankel Determinant of Certain Subclass of Bi-Univalent Functions.

In this study, we introduce a novel subclass of bi-univalent functions, which are of considerable interest in various fields of mathematics, including complex analysis and geometric function theory. By employing the property of subordination, we define these bi-univalent functions as R(t, γ, λ) and impose constraints on the coefficients | an |. Our investigation provides the upper bounds for the bi-univalent functions in this newly developed subclass, specifically for n=2, 3, 4, and 5. We then derive the third Hankel determinant for this particular class, which reveals several intriguing scenarios. These findings 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. [ABSTRACT FROM AUTHOR]