Regarding an analytic function

Question

Let $f$ be an analytic function on the closed unit disc. Let $f(0)=0$ and $f’(0)\neq 0$. Does this mean that $\sup\Re(f)>0$?

Answer 1

With $f(0)=0$, open mapping theorem shows $|z|<\epsilon$ has an open neighborhood in it's image, have some points $w$ with $\Re w>0$.