Let $D=D(0,1)$ and let $f$ be a holomorphic function on $D$ with $f'$ bounded on $D$.
Let $g=(1-|z|) |f'(z)|$
I need to use the maximum modulus principle on $g$ to prove that it's maximum is reached within $D$ but I don't manage to prove that $g$ is holomorphic.
Is that even true that g is holomorphic?
Any help or hint is much appreciated.