Bounding injective holomorphic mappings on $\mathbb{C}^n$ in the spirit of Andersen-Lempert

Question

I'm hoping the following is true.

Let $Aut_0^I(\mathbb{C}^n)$ denote the set of holomorphic automorphisms $\phi:\mathbb{C}^n \to \mathbb{C}^n$ s.t. $\phi(0)=0$ and $d \phi(0) = I_n$ where $I_n$ is the identity matrix. Let $F:\mathbb{C}^n \to \mathbb{C}^n$ be an injective holomorphic mapping s.t. $F(0) = 0$ and $d F(0) = I_n$. Let $D>0, \epsilon >0$, then there exists a $\phi \in Aut_0^I(\mathbb{C}^n)$ s.t. $\sup_{||x|| = D} ||F\circ \phi (x)|| < D + \epsilon$