I am hoping the following is true. Mention of related ideas/topics are appreciated.



> 
> Suppose $F:\mathbb{C}^n \to \mathbb{C}^n$ is a injective holomorphic mapping such that $F(0)=0$ and $dF(0) = I\_n$ where $I\_n$ is the $n \times n$ identity matrix. Let $\partial B$ denote the boundary of the unit ball centered at the origin in $\mathbb{C}^n$. Let $M = \sup\_{x \in \partial B} ||F(x)||$ where $|| \cdot ||$ is the usual Euclidean norm. Then $\partial B \cap \{x: M = ||F(x)|| \}$ is equal to one of three things:
>  i) $\{p\}$ for some point $p$,
>  ii) $\{\alpha p : |\alpha|=1 \}$ for some point $p$, 
>  iii) $\partial B$
> 
> 
> 



About maxima of injective holomorphic maps on $\mathbb{C}^n$
This is not true.


Consider $(z,w) \mapsto (z+w^2, w)$.


Let $(p\_1,p\_2) = p \in \partial B$ s.t. $||F(p)|| = M$.
Then the points $(\alpha^2 p\_1, \alpha p\_2)$ with $|\alpha | = 1$ are also in $\partial B$ and $\{x: M = ||F(x)||\}$.


