I have to decide if this family is normal or not
$\mathscr{F}=\{f\in\mathscr{H}(D(0,1)): f(0)=2i\ \&\ \Im f(z)>0\ \forall z\in D(0,1)\}$
where $\Im f(z)$ is the imaginary part of $f(z)$.
I am trying to use Montel's theorem but I don't know how to decide if the family is bounded. I also tried to use a Möbius transformation but I didn't conclude anything. Can somebody help me? Thank you very much!