Summary: Let \(E\) denote the unit disc in \(\mathbb{C}\). For any domain \(G\subset\mathbb{C}^n\) define \[ g_D(a,z):= \inf_{\substack{ \varphi\in{\mathcal O}(E,D)\\ \varphi(0)=z\\ a\in\varphi(E)}} \Biggl\{ \prod_{\lambda\in \varphi^{-1}(a)} |\lambda |^{\text{ord}_\lambda (\varphi-a)}\Biggr\}, \qquad a,z\in D, \] where \({\mathcal O}(E,D)\) denotes the set of all holomorphic mappings \(E\to D\) and \(\text{ord}_\lambda (\varphi-a)\) denotes multiplicity of \(\varphi-a\) at \(\lambda\). The function \(g_D\) is called the pluricomplex Green function for \(D\).
The author proves that the pluricomplex Green function has the product property \(g_{D_1\times D_2}= \max\{g_{D_1}, g_{D_2}\}\) for any domains \(D_1\subset \mathbb{C}^n\) and \(D_2\subset \mathbb{C}^m\).