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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) $\partial B$

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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 $\{x: M = ||F(x)||\}$.

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