Let $(P,H)$ be a Euclidean Hartogs figure in $\mathbb{C}^n$ and $f:H\to \mathbb{C}^n$ a holomorphic injective map, then we know that $f$ extend holomorphically to polidisc $P$ (i.e. there is a holomorphic map $F:P\to\mathbb{C}^n$ such that $F\equiv f$ on $H$), is true that $F$ is also injective?
Any hint would be appreciated.