In $\mathbb{C}^n,\ n\geq 2$, there is no bijection between unit disk $B^n(0,1)$ and unit polydisk $P^n(0,1)$. But if we wish to find injective holomorphic mapping from unit disk to polydisk(whose image contains origin), inclusion is the obvious mapping or suitable automorphisms of unit disk(which is sort of inclusion, after applying automorphism). But can we go beyond unit disk in polydisk, by means of injective holomorphic mapping. Are there results in this direction available?



injective holomorphic mapping between unit disk and unit polydisk
Take $f(z\_1,z\_2):=(z\_1,sz\_2+(1-s)z\_1)$, $0<s<1$. For example, $s=1/2$. Then $f$ is an injective holomorphic mapping from $B(0,1)$ to $P(0,1)$, ($n=2$), sending $0$ to $0$. Moreover, $\|f(s,0)\|>1$ for $s<1$, near $1$, i.e. $f(s,0)\notin B(0,1)$..


