AI 中文总结
研究复Banach空间开单位球上B型归一化β-螺形拟凸映射类的二阶Hankel行列式，消除映射形式的限制性假设，得到β=0时的严格尖锐上界，提出β≠0时的未解决问题。
AI 中文摘要
我们针对复Banach空间开单位球𝔹上的B型归一化β-螺形拟凸映射类𝒞_B^β(𝔹)，建立了二阶Hankel行列式H_{2,2}(F)=A_2 A_4 - A_3^2的界。通过利用基于方向切片齐次多项式展开的广义框架，我们消除了映射具有F(x)=g(x)x形式的标准限制性假设。在这些更弱的操作条件下，我们通过经典Carathéodory泛函参数对目标标量不变量A_n进行参数化。严格的优化分析证明，对于经典非螺形情况β=0，所建立的上界是严格尖锐的，其最大值为1/8。通过构造显式的多维极值映射（提升了对应的单变量凸轮廓）验证了该尖锐界。最后，提出了关于β≠0时精确变分行为的未解决公开问题。
英文摘要
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 $\mathcal{C}_{B}^β(\mathbb{B})$ of normalized $β$-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 $β= 0$, 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 $β\neq 0$ is formulated.
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