Conformal map from upper-half disk to disk

Question

I am trying to find a conformal mapping $f$ from the upper half-disk $\{z:|z|<1 \text{ and } \text{Im}(z)>0\}$ to the whole disk that maps $\{-1,0,1\}$ onto $\{-1,-i,1\}$.
The hint that I am given is to try to find Mobius mappings $A,B$ so that $f=A \circ S \circ B$, where $S(z)=z^2$.
I'm completely stuck with this, and I feel like it should be a lot easier than I'm making it. Any tips?

Answer 1

Hint. $$ A(z)=\frac{z-1}{z+1} $$ maps the upper semicircle to the first quadrant.

If $S(z)=z^2$, the $S\circ A$ maps the upper semicircle to the upper half plane.