Continuous $f:\overline{\mathbb{D}} \to \mathbb{C}$ such that $f$ is holomorphic in $\mathbb{D}$ and $|f(z)|=1, \forall z \in \partial\mathbb{D} \ ?$ Could anyone advise me how to find all continuous $f:\overline{\mathbb{D}} \to \mathbb{C}$ such that $f$ is holomorphic in $\mathbb{D}$ and $|f(z)|=1, \forall z \in \partial\mathbb{D} \ ?$ Do I use Riemann mapping theorem?
Thank you. 
 A: First use the maximum modulus principle to conclude that $f:\mathbb{D}\to\mathbb{D}$.
Then study the case in which $f$ doesn't vanish. If it doesn't vanish then $1/f$ is holomorphic. This implies that either $|f|$ or $|1/f|$ has an interior maximum or they are constant. Therefore they are constant.
Then the idea is to write $f$ as a product of $f_{z_0,n}$ where $f_{z_0,n}:\mathbb{D}\to\mathbb{D}$ and it only vanishes at $z_0$ to order $n$.
To describe $f_{z_0,n}$ it is enough to describe $f_{z_0}:=f_{z_0,1}$.
By a Moebius transformation (you must compute) we can take $z_0$ to the origin.
Then use Schwarz's lemma to conclude that $|f_{0}(z)|\leq |z|$. Therefore $f_{0}(z)/z$ is holomorphic and goes from $\mathbb{D}$ to itself. But $f_{0}(z)/z$ doesn't vanish anymore, so by the argument at the beginning it must be constant. Therefore $f_{0}(z)=cz$. This allows you to write down all the $f_{z_0}$. An then all the $f$.
To check that you got the $f_{z_0}$ look here at the article on Blaschke functions. The $B(a,z)$ in the article would be out $f_{a,1}(z)$.
So, the answer is going to be a finite Blaschke product. We don't want infinite Blaschke product because we don't want zeros accumulating in the boundary of $\mathbb{D}$.
