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) $\partial B$