与月牙域和豆形域相关的高阶Schippers泛函的精确界
Sharp Bounds for Higher-Order Schippers Functionals Associated with Lune and Bean Domains
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中文总结 AI 辅助
该研究针对与月牙域、豆形域相关的单叶函数子类，推导了三阶、四阶Schippers泛函的精确界，确定了极值函数与像域几何，还得到了初始Grunsky系数的精确估计。
中文摘要 AI 辅助
我们获得了与非经典几何域相关的单叶函数子类的三阶和四阶Schippers泛函$|\sigma_3(f)(0)|$和$|\sigma_4(f)(0)|$的精确界。具体而言，我们研究了由从属关系$\frac{zf'(z)}{f(z)} \prec z+\sqrt{1+z^2}$确定的月牙星形类$\mathcal{S}_{\leftmoon}^*$、由$1+\frac{zf''(z)}{f'(z)} \prec z+\sqrt{1+z^2}$确定的月牙凸类$\mathcal{C}_{\leftmoon}$，以及与$\mathfrak{B}(z)=\sqrt{1+\tanh z}$相关的豆形域类$\mathcal{BT}_{\mathfrak{B}}$。利用Carathéodory系数参数化和极值优化技术，我们推导了原点处高阶施瓦茨导数的精确估计，并确定了相应的极值函数。此外，还提供了相关极值像域的几何描述以说明尖锐性现象。所得结果进一步给出了初始Grunsky系数$g_{1,1}$和$g_{1,2}$的精确界，这些发现为与月牙形和豆形域相关的单叶函数的高阶施瓦茨结构提供了精确描述。
英文摘要
We obtain sharp bounds for the third- and fourth-order Schippers functionals, $|σ_3(f)(0)|$ and $|σ_4(f)(0)|$, for subclasses of univalent functions associated with non-classical geometric domains. In particular, we investigate the lune-starlike class $\mathcal{S}_{\leftmoon}^*$ and the lune-convex class $\mathcal{C}_{\leftmoon}$ determined by the subordination \[ \frac{zf'(z)}{f(z)} \prec z+\sqrt{1+z^2}, \qquad 1+\frac{zf''(z)}{f'(z)} \prec z+\sqrt{1+z^2}, \] respectively, together with the bean-domain class $\mathcal{BT}_{\mathfrak{B}}$ associated with \[ \mathfrak{B}(z)=\sqrt{1+\tanh z}. \] Using Carathéodory coefficient parametrizations and extremal optimization techniques, we derive exact estimates for the higher-order Schwarzian derivatives at the origin and identify the corresponding extremal functions. In addition, geometric descriptions of the associated extremal image domains are provided to illustrate the sharpness phenomena. The obtained results further yield sharp bounds for the initial Grunsky coefficients $g_{1,1}$ and $g_{1,2}$. These findings provide a precise description of higher-order Schwarzian structures for univalent functions related to lune and bean shaped domains.
