First, let me define a notation of $H(G_1\times G_2 \times \ldots \times G_m)$. We say that $f\in H(G_1\times G_2 \times \ldots \times G_m)$ if $$f:G_1\times G_2 \times \ldots \times G_m \rightarrow \mathbb{C}$$ and any function $f(x_1,x_2,\ldots,x_k, \ldots, x_m)$ where $x_1,x_2,\ldots,x_{k-1},x_{k+1},\ldots,x_m$ are fixed in $G_1,G_2,\ldots,G_{k-1},G_{k+1},\ldots,G_m$ is holomorphic on $G_k.$
I am so curious. Is there analogue of Hurwitz's theorem for a system of functions?
What I mean is the following:
Let $G_1,G_2,\ldots,G_m$ be regions in $\mathbb{C}$ and suppose that any sequences $\{f_{n,k}\}_{n\in \mathbb{N}}$ in $H(G_1\times G_2 \times \ldots G_m)$ converge to $f_k$. If for all $k\in\{1,2,\ldots,m\},$ $f_k\not\equiv 0,$ $\overline{B}(a_k;R_k)\subset G_k,$ $f_k(z)\not=0$ for $|z-a_k|=R_k,$ and there exists $(y_1,y_2,\ldots,y_m)\in \overline{B}(a_1;R_1)\times \overline{B}(a_2;R_2) \times \ldots \overline{B}(a_m;R_m)$ such that $$(f_1(y_1,y_2,\ldots,y_m),f_2(y_1,y_2,\ldots,y_m),\ldots,f_m(y_1,y_2,\ldots,y_m))=(0,0,\ldots,0),$$ then there is an integer $N$ such that for $n \geq N$ there exists $(y_{n,1},y_{n,2},\ldots,y_{n,m})\in \overline{B}(a_1;R_1)\times \overline{B}(a_2;R_2) \times \ldots \overline{B}(a_m;R_m)$ such that $$(f_{n,1}(y_{n,1},y_{n,2},\ldots,y_{n,m}),f_{n,2}(y_{n,1},y_{n,2},\ldots,y_{n,m}),\ldots,f_{n,m}(y_{n,1},y_{n,2},\ldots,y_{n,m}))=(0,0,\ldots,0).$$ Moreover, $y_{n,k}\rightarrow y_k$ as $n \rightarrow \infty$ for all $k\in\{1,2,\ldots,m\}$.
Could you point out the references for the theorem or give me some ideas of the proof? Thank you so much for your help.
Masih
