AI 中文总结
本文针对归一化反正切映射生成的Ma-Minda凸类，利用从属技巧等开展系数分析，得到泰勒系数、Hankel行列式等的精确估计，确定对应极值函数，证明所得不等式为最优。
AI 中文摘要
令$\boldsymbol{\textit{C}}_{\boldsymbol{\textit{arcsin}}}$表示由归一化反正弦映射$\boldsymbol{\textit{φ}}(\boldsymbol{\textit{z}})=1+\frac{2}{\boldsymbol{\textit{π}}}\boldsymbol{\textit{arcsin}}\boldsymbol{\textit{z}}$生成的Ma-Minda凸函数子类。针对该函数族，本文利用从属技巧、Carathéodory函数及Schwarz函数的精确估计开展统一系数分析，由此推导初始泰勒系数、对数系数及对数系数与逆对数系数间某些差值的精确估计；进一步确定二阶Hankel行列式$\boldsymbol{\textit{H}}_{2,2}(\boldsymbol{\textit{f}})$，以及与函数及其逆的对数系数相关的Hankel行列式$\boldsymbol{\textit{H}}_{2,1}(\boldsymbol{\textit{F}}_{\boldsymbol{\textit{f}}}/2)$、$\boldsymbol{\textit{H}}_{2,1}(\boldsymbol{\textit{F}}_{\boldsymbol{\textit{f}}^{-1}}/2)$的精确界；还得到初始广义Zalcman泛函和广义Fekete–Szegö泛函的精确估计，并识别出对应极值函数，证明所有所得不等式均为最优。
英文摘要
Let $\mathcal{C}_{\arcsin}$ denote the Ma--Minda subclass of convex functions generated by the normalized arcsine mapping $φ(z)=1+\frac{2}π\arcsin z.$ For this family, we develop a unified coefficient analysis based on subordination techniques, Carathéodory functions and sharp estimates for Schwarz functions. As consequences, we derive sharp estimates for the initial Taylor coefficients, logarithmic coefficients and certain differences involving the logarithmic and inverse logarithmic coefficients. We further determine the exact bounds for the second Hankel determinant $H_{2,2}(f)$ together with the Hankel determinants $H_{2,1}(F_f/2)$ and $H_{2,1}(F_{f^{-1}}/2)$ associated with the logarithmic coefficients of a function and its inverse. Moreover, sharp estimates are obtained for the initial generalized Zalcman functional and the generalized Fekete--Szegö functional. In every case, the corresponding extremal functions are identified, showing that all of the obtained inequalities are best possible.
