复Banach空间中\(\mathcal{S}_\mathbb{B}^{*}(α)\)的Hankel、Fekete - Szegö和Zalcman泛函的精确估计
Sharp Estimates for Hankel, Fekete-Szegö and Zalcman Functionals for $\mathcal{S}_\mathbb{B}^{*}(α)$ in Complex Banach Spaces
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中文总结 AI 辅助
研究复Banach空间单位球上\(\alpha\)阶星形映射相关问题，利用弗雷歇导数和辅助引理建立二阶汉克尔行列式等泛函的精确上界，证明其精确性，还将相关经典结果扩展到复Banach空间。
中文摘要 AI 辅助
受Cho等人[CKKLS2018]关于单位圆盘中\(\alpha\)阶星形函数的精确系数估计的启发，我们研究了复Banach空间单位球上定义的\(\alpha\)阶星形映射的相应问题。利用弗雷歇导数和合适的辅助引理，我们建立了与这类映射相关的二阶汉克尔行列式、费克特 - 塞戈泛函和扎尔茨曼泛函的精确上界。通过确定相应的极值映射，证明了每种情况下得到的估计都是精确的。此外，当基础Banach空间为复平面时，我们的结果简化为已知的一维精确估计，从而将Cho等人[CKKLS2018]的几个经典结果扩展到复Banach空间的情形。
英文摘要
Inspired by the sharp coefficient estimates established by Cho \emph{et al.}\cite{CKKLS2018} for starlike functions of order $α$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $α$ defined on the unit ball of a complex Banach space. Employing Fréchet derivatives together with suitable auxiliary lemmas, we establish sharp upper bounds for the second-order Hankel determinant, the Fekete--Szegö functional, and the Zalcman functional associated with this class of mappings. In each case, the obtained estimates are shown to be sharp by identifying the corresponding extremal mappings. Furthermore, our results reduce to the known one-dimensional sharp estimates when the underlying Banach space is the complex plane, thereby extending several classical results of Cho \emph{et al.} \cite{CKKLS2018} to the setting of complex Banach spaces.
