Sharp Bohr-Rogosinski radii for Schwarz functions and Euler operators in C^n

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc ${\mathbb{D}}^n$. We establish a sharp extension of the classical Bohr inequality, proving that the Bohr radius remains $R_n=1\mathrm{/}(3n)$ for the family of holomorphic functions bounded by unity in the multivariate setting. Further, we provide a definitive resolution to the Bohr-Rogosinski phenomenon in several complex variables by determining sharp radii for functional power series involving the class of Schwarz functions ${\omega }_{n,m}\in {\mathcal{B}}_{n,m}$ and the local modulus $\mathrm{\mid }f(z)\mathrm{\mid }$. By employing the radial (Euler) derivative operator $Df(z)={\sum }_{k=1}^n{z}_k\frac{\mathrm{\partial }f(z)}{\mathrm{\partial }{z}_k}$, we obtain refined growth estimates for derivatives that generalize well-known univariate results to ${\mathbb{C}}^n$. Finally, a multidimensional version of the area-based Bohr inequality is established. The optimality of the obtained constants is rigorously verified, demonstrating that all established radii are sharp.