Let $f : \overline{U} → C$ is holomorphic where $|f(z)| = 1$ for $|z| = 1$. Show that if $f$ has no zeros in $U$, then $f$ must be a constant. (Use maximum modulus principle). Where $U$ is open unit disk.
I checked that this statememt was proved on the unit disk, but I couldn't prove that $f$ is a constant in $U$. Any help would be really appreciated.
By the maximum modulus principle, we have $$ \sup_{z\in U} |f(z)|\leq 1.$$ Since $f$ has no zero in $U$, it follows that $1/f$ is holomorphic in $U$. Moreover, $1/|f|=1$ on $\partial U$. Hence it follows from the maximum modulus principle that $$ \sup_{z\in U} (1/|f|) \leq 1.$$ Hence $|f(z)|=1$ for all $z\in U$. Since $U$ is a domain, $f$ is constant (see link).
