What is an example an injective holomorphic function whose derivative is zero at a point?

Question

Let $G$ be open in $\mathbb{C}$.

Let $f:G\rightarrow \mathbb{C}$ be an injective holomorphic function.

Is it possible that at some point $z_0\in G$, $f'(z_0)=0$?

Answer 1

It is not possible. I am using the following theorem:

Let $f$ be analytic at $z_0$.Suppose $f(z_0)=w_0$. Assume that $f(z)-w_0$ has a zero of order $n$ at $z=z_0$. Then there is a $\epsilon \gt 0$ such that for any $0 \lt \delta \lt \epsilon$, there is a $r \gt 0$ such that for $a \in B(w_0,r)$, $f(z)-a$ has exactly $n$ distinct solutions in $B(z_0,\delta)$.

Suppose that $f'(z_0)=0$ for some $z_0 \in G$. Conisder $g(z)=f(z)-f(z_0)$. Then $g(z_0)=0$ and $g'(z_0)=0$. Thus $z_0$ is a zero of order $2$. By above theorem we reach a contradiction.