Let $U$ be a simply connected domain in $\mathbb{C}$, let $w\in U$. If $f,g\in Aut(U)$ such that $f(w)=g(w)$ and $f'(w)=g'(w)$, show f=g.

Question

I do not even know how to approach this problem when $U=D(0,1)$ or $\mathbb{C}$.

Thank you!