On Hankel Determinants of $\beta$-Spirallike Mappings in Complex Banach Spaces

This paper is devoted to the study of the second Hankel determinant for normalized $\beta$-spirallike mappings on the unit ball of a complex Banach space. Using the relationship between Banach-space holomorphic mappings and corresponding one-variable $\beta$-spirallike functions, we first establish sharp estimates for the initial homogeneous expansion coefficients. These coefficient inequalities are subsequently applied to derive a sharp bound for the second Hankel determinant.