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.
Let $\phi:D\to D$ be defined by $\phi(z)=\frac{z_0-z}{1-\overline{z_0}z}$, so that $\phi$ is a bijective analytic map such that $\phi(z_0)=0$ and $\phi(\phi(z))=z$ for all $z\in D$. Let $g=\phi\circ f\circ \phi$. Note that $g^n=\phi\circ f^n\circ\phi$ and $f^n=\phi\circ g^n\circ \phi$ (where the exponents denote iteration).
Since $g:D\to D$ is analytic, $g(0)=0$, and $|g'(0)|<1$, Schwarz's Lemma implies that $|g(z)|<|z|$ for all $z\in D\setminus\{0\}$. It follows that if $0<r<1$, then there exists $c$ with $0<c<1$ such that $|g(z)|\leq c|z|$ for all $z$ with $|z|\leq r$ (using compactness of the closed disk of radius $r$). Hence, $|g^n(z)|\leq c^n|z|$ for all $z$ with $|z|\leq r$. This implies that $g^n\to 0$ uniformly on compact sets, which in turn implies that $f^n=\phi\circ g^n\circ\phi\to z_0$ uniformly on compact sets.