Let $D=D(0,1)$.
Let $f \in \mathcal{O}(D)$ be holomorphic such that $f'$ is bounded in $D$.
let $g=(1-|z|) |f'(z)|$.
I try to prove that $g$ reaches its maximum in $D$.
$g$ is continuous (holomorphic) and bounded , but is it enough to prove that it reaches a maximum in $D$?
Thank you!