Entire functions of several complex variables with the same modulus

Question

The $1D$ version of the following question was rated as inappropriate for Mathoverflow. Let $f,g$ be entire functions of $n$ complex variables such that $$ \forall z\in \mathbb C^n,\quad \vert f(z)\vert= \vert g(z)\vert. $$ Does that imply that $g=cf$ with $c\in \mathbb C, \vert c\vert=1$?

Answer 1

I suppose that $n=2$, to give the idea of the proof. Let $a=(a_1,a_2)$ a point (fixed) such that $g(a)\not =0$, put $c=\frac{f(a)}{g(a)}$. Let $b=(b_1,b_2)$ another point. Let $\phi(t)=f((1-t)a+tb)=f((1-t)a_1+tb_1,(1-t)a_2+tb_2)$, and $\psi(t)=g((1-t)a+tb)$. Then $\phi$ and $\psi$ are entire functions of the complex variable $t$, and we have $|\psi(t)|=|\phi(t)|$ for all $t$. By the one-dimension case, there exists $c_1$ such that $\phi(t)=c_1\psi(t)$ for all $t$. For $t=0$, we get $c_1=c$ and for $t=1$, that $f(b)=cg(b)$, and we are done.