# About maxima of injective holomorphic maps on $\mathbb{C}^n$

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) $$\partial B$$

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