fixed point of a holomorphic function on a disk

Let $\mathbb{D} = \{ z : |z|<1 \}$ and $f$ an holomorphic function on $\mathbb{D}$ and continuous on $\overline{\mathbb{D}}$ such that $f(\overline{\mathbb{D}}) \subset \mathbb{D}$.

Prove the following:

1. There exists single point $z^* \in \mathbb{D}$ such that $f(z^*)=z^*$ (obvious by Rouche theorem).
2. Let $f_1=f,...,f_{n+1}=f(f_n)$ show that $f_n(z) \longrightarrow z^*$ uniformly.
• Since this is homework, you should explain your own effort to solve this problem. That alone may give you the last hint needed to answer it yourself. – hardmath Aug 2 '12 at 15:38
• How is Rouche's theorem used for 1? I can see a contraction, but not Rouche... – copper.hat Aug 2 '12 at 15:46
• @copper.hat: I guess the idea is that $\lvert f(z) \rvert \lt \lvert -z\rvert$, so $z \mapsto -z$ and $z \mapsto f(z) - z$ have the same numbers of roots in $\mathbb{D}$ (but this would need $f$ to be holomorphic in a neighborhood of the closed disk, I believe). – t.b. Aug 2 '12 at 15:53
• Or at least a further word than "obvious" should be said... – t.b. Aug 2 '12 at 15:59
• @GunnarMagnusson, the map $f(z)=z/2$ is constant? – JSchlather Aug 2 '12 at 17:54

The key to all this is that $f(\bar{D}) \subset D$:

1. Since $f(\bar{D})$ is compact, there exists $r_0>0$ such that $f(\bar{D})\subset D_{r_0}$. So for any $r_0<r<1$ we have $|f(z)|=|(f(z)-z)+z|<|-z|$ on $D_r$ so by Rouché's theorem $-z$ and $f(z)-z$ have the same zeroes, which is one. Since this is valid for any $r>r_0$ the uniqueness result follows.

2. Since $|f(z)/z|=|f(z)|$ is continuous on $\partial D$ it has a maximum $M$, and by hypothesis $M<1$. So, assuming for the moment that $f(0)=0$, by the maximum principle we get $|f(z)|\leq M|z|$ for $z\in D$. This gives that $|f_n(z)|\leq M|f_{n-1}(z)|$ in $D$, and so $|f_n(z)|\leq M^n|z|$. Taking supremums over $\bar{D}$ and then the limit as $n\to \infty$ the result follows.

Assume now that $f(0)\neq 0$ then everything just said applies to $g=h\circ f\circ h^{-1}$ (with $h$ an appropiate automorphism of the disk), and the result follows in general.