I have to find a biholomorphic map from $\{z \in \mathbb{c} : |z|<1, $Im$ z>0\}$ to the unit disk.
My idea: If I can somehow map the upper half disk into the upper half plane (biholomorphically) I am done, since I can then compose it with Caley transform to get into the unit disk. If I use $f(z)=1/z$ it will only map it to outside of the upper half disk in the half plane, so can I edit it a bit to capture the interior of half disk as well?
First, $$ f_1(z)=\frac{1+z}{1-z} $$ takes the upper half of the unit disk to the first quadrant.
Then $$ f_2(z)=z^2 $$ takes the first quadrant to the upper half plane.
Finally, $$ f_3(z)=\frac{z-i}{z+i} $$ maps the upper half plane to the unit disk.
Altogether $$ f(z)=(f_3\circ f_2\circ f_1)(z)= \frac{\left(\frac{1+z}{1-z}\right)^2-i}{\left(\frac{1+z}{1-z}\right)^2+i}= -i\cdot\frac{z^2+2iz+1}{z^2-2iz+1}. $$