Third-Order Hankel Determinant for a Class of Bi-Univalent Functions Associated with Sine Function

This paper investigates a new subclass of bi-univalent analytic functions defined on the open unit disk in the complex plane, associated with the subordination to 1 + sinz. Coefficient bounds are obtained for the initial Taylor?Maclaurin coefficients, with a particular focus on the second- and third-order Hankel determinants. To illustrate the non-emptiness of the proposed class, we consider the function ? 1 + tanh z, which maps the unit disk onto a bean-shaped domain. This function satisfies the required subordination condition and hence serves as an explicit member of the class. A graphical depiction of the image domain is provided to highlight its geometric characteristics. The results obtained in this work confirm that the class under study is non-trivial and possesses rich geometric structure, making it suitable for further development in the theory of geometric function classes and coefficient estimation problems.