Schwarz函数和方向导数算子在复空间C^n中的锐Bohr半径
Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in \mathbb{C}^n
AI总结：
本文通过Schwarz函数和方向导数算子在复空间C^n中研究多变量复变量的锐Bohr半径，推导出导数增长估计并证明所获常数的最优性。
摘要：
本文致力于研究复空间C^n上有界全纯映射的多维精炼Bohr型不等式的类比。我们通过确定涉及Schwarz函数ω_{n,m}∈B_{n,m}和局部模|f(z)|的功能幂级数的锐半径，为多变量复变量中的Bohr现象提供确定性解决方案。通过运用方向导数算子∂_u f(z)=∑_{k=1}^{n} u_k ∂f(z)/∂z_k，其中u=(u_1,u_2,…,u_n)∈C^n且|u_1|+|u_2|+…+|u_n|=1，我们获得了导数的细化增长估计，将已知的单变量结果推广到C^n。所获得的常数的最优性得到了严格验证，证明所有确定的半径都是锐的。
英文摘要：
This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}Δ(0;1_n)$. We provide a definitive resolution to the Bohr phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions $ω_{n,m}\in\mathcal{B}_{n,m}$ and the local modulus $|f(z)|$. By employing the directional derivative operator $\partial_uf(z) = \sum_{k=1}^{n} u_k \frac{\partial f(z)}{\partial z_k}$, where $u=(u_1,u_2,\ldots,u_n)\in\mathbb{C}^n$ such that $|u_1|+|u_2|+\ldots+|u_n|=1$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to $\mathbb{C}^n$. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.
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