Let $f = u + iv$ be a holomorphic function on unit (open) disk $\mathbb D^2$.
and Suppose $v$ is nonnegative and $v(0)=0$.
Can we determine all such $f$?
I wanted to use schwarz lemma, but I couldn't catch anything about bound condition.
I think it is about liouville's theorem.
Is it related to harmonic function?