One of the past qualifying question is as follows,
Let $D$ be a bounded domain, and f analytic from $D$ on $D$.
Let $z_{0}$ be a fixed point of $f$ and $|f^`(z_{0})|< 1$.
Now I am trying to show that nth iterate of $f$ converges uniformly on compact subsets of D to $z_{0.}$
I have a hunch that it is an application of Mittag- Leffler theorem. But this is getting nowhere with that thinking. I would love to see rigorous proof of this problem.