Find conformal mapping that maps set $\{(x,y)\in\mathbb{R}^2 : \operatorname{y\le 0}\}$ to unit ball. Find conformal mapping that maps set $A=\{(x,y)\in\mathbb{R}^2 :  \operatorname{y\le 0}\}$ to unit disk. I know that such a mapping exists from Riemann Theorem.
Note: I don't want full answer. I expect only some starting point hint.
Thanks
Edit:
I can use rotation by $\frac{-\pi}{2}$ on map:
$$ z \mapsto \frac{z-1}{z+1}$$
So my map looks like that: $$z \mapsto e^{-i\frac{\pi}{2}} \frac{z-1}{z+1}$$
Edit 2:
$$z \mapsto  \frac{e^{-i\frac{\pi }{2}}z-1}{e^{-i\frac{\pi }{2}}z+1}$$

A map from the complex half-plane of numbers with positive real part to the unit disc is given by $$ z \mapsto \frac{z-1}{z+1}$$
For details on this see an earlier question Mapping half-plane to unit disk?
You should be able to modify this for you purpose; note that your set corresponds to the  half-plane with negative imaginary part. 
Your attempt is good in that you can indeed compose this map with a rotation.  However, note that you have to rotate first and then use the map given above. (You did it the other way round.) 
