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., $f^{\mathrm{♯}}(z)\le C\underset{j}{max}(1-|z_j{|}^2{)}^{-1}$ for some constant $C>0$.