Let $f$ be an analytic map defined on $D=\{z:|z|<1\}$ such that $|f(z)|\le 1\forall z\in D$. Then which of the following statements are true?
There exists $z_0\in D$ such that $f(z_0)=1$.
The image of $f$ is an open set.
$f(0)=0$
$f$ is constant.
Well first I wonder how can $|f|$ assume the value $1$? 4 is correct I guess, and 2 may be correct by the Open Mapping Theorem, but I am not able to figure out the other options.