Let $h(z) = g(f(z))$. If $f$ and $h$ are non-constant holomorphic function on domains in $\mathbb C^n$, then is $g$ holomorphic?

Question

Let $h(z) = g(f(z))$. If $f$ and $h$ are non-constant holomorphic function on domains in $\mathbb C^n,\, n>1$, then is $g$ holomorphic? I have seen various discussion here, however most proofs relies on one variable techniques! It would be really nice to have some references.