a) Prove that the most general $1-1$ conformal map of the upper half-plane onto itself is of the form $$z \to \frac{az+b}{cz+d}$$ where $a,b,c,d \in \mathbb{R}$ and $ad-bc =1$.
b) Let $f$ be a $1-1$ analytic function from the plane to itself. What can it be? A full explanation from first principles is wanted.
Edit: I think I have a decent proof of part (a), so any hints for part (b) are welcome.
Thanks,