Second Hankel Determinant for $\beta$-Spirallike Convex Mappings in Complex Banach Spaces

Abstract

We establish the bound for the second-order Hankel determinant $H_{2,2}(F)=A_2{A}_4-A_3^2$ associated with the class ${\mathcal{C}}_B^{\beta }(\mathbb{B})$ of normalized $\beta$-spirallike quasi-convex mappings of type $B$ on the open unit ball $\mathbb{B}$ of a complex Banach space. By utilizing a generalized framework based on a directional slice homogeneous polynomial expansion, we eliminate the standard, restrictive assumption that the mapping is of the form $F(x)=g(x)x$. Under these weaker operational conditions, we parameterize the targeted scalar invariants $A_n$ via the classical Carathéodory functional parameters. A rigorous optimization analysis proves that the established upper bound is strictly sharp for the classical non-spirallike case $\beta =0$, yielding a maximal value of $1/8$. This sharp bound is verified by constructing explicit multi-dimensional extremal mappings that lift the corresponding single-variable convex profile. Finally, an unresolved open question regarding the exact variational behavior for $\beta \ne 0$ is formulated.

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