Sharp Bohr Radii for Schwarz Functions and Directional derivative Operators in $\mathbb{C}$^n

This paper is devoted to the investigation of multidimensional analogues of refined Bohr-type inequalities for bounded holomorphic mappings on the unit polydisc $\mathbb{P}\mathrm{\Delta }(0;1_n)$. 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 ${\omega }_{n,m}\in {\mathcal{B}}_{n,m}$ and the local modulus $\mathrm{\mid }f(z)\mathrm{\mid }$. By employing the directional derivative operator ${\mathrm{\partial }}_{u}f(z)={\sum }_{k=1}^n{u}_k\frac{\mathrm{\partial }f(z)}{\mathrm{\partial }{z}_k}$, where $u=(u_1,u_2,\dots ,u_n)\in {\mathbb{C}}^n$ such that $\mathrm{\mid }{u}_1\mathrm{\mid }+\mathrm{\mid }{u}_2\mathrm{\mid }+\dots +\mathrm{\mid }{u}_n\mathrm{\mid }=1$, 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.