Normal criterion on the polydisc in $ \mathbb{C}^n $

This paper established a slice characterization for normal families of holomorphic functions on the unit polydisc $ \mathbb{D}^n\subset\mathbb{C}^n $. The main result (Theorem 2.2) showed that a family $ \mathcal{F}\subset\mathcal{O}(\mathbb{D}^n) $ was normal if, and only if, for every point $ a\in\mathbb{D}^n $ and every coordinate direction $ 1\le j\le n $, the corresponding one-dimensional slice family $ \mathcal{F}_{a, j} $ was normal on the unit disc. Building on this characterization, we introduced the notion of normal functions on $ \mathbb{D}^n $ and proved a metric criterion (Proposition 2.7): A function $ f\in\mathcal{O}(\mathbb{D}^n) $ was normal exactly when its spherical derivative grew at most as fast as the Poincaré metric density, i.e., $ f^\sharp(z)\le C\max_j(1-|z_j|^2)^{-1} $ for some constant $ C > 0 $.