Finding the image of the Mobius transformation ( upper half plane to unit disk )

Exercise:

Find the Mobius transformation T that maps $\left\{ 0,1,\infty  \right\}$ to $\left\{ 1,i,-1 \right\}$

Solution: we know that $T\left( z \right)=\frac{az+b}{cz+d}\,\,\,\,a,b,c,d\in \mathbb{C}\,\,:\,\,ad-bc\ne 0$ is a Mobius transformation

but $T\left( \infty  \right)\ne \infty $ so $c\ne 0$ thus $c=1$ hence $T\left( z \right)=\frac{az+b}{z+d}$

also $T\left( \infty  \right)=-1$ so $a=-1$ hence $T\left( z \right)=\frac{-z+b}{z+d}$

but $T\left( 0 \right)=1\Leftrightarrow \frac{b}{d}=1\Rightarrow b=d$ hence $T\left( z \right)=\frac{-z+b}{z+b}$

notice that $T\left( 1 \right)=i\Leftrightarrow \frac{-1+b}{1+b}=i\Leftrightarrow -1+b=i+ib\Leftrightarrow b=\frac{-1-i}{-1+i}=i$

therefore, $T\left( z \right)=\frac{-z+i}{z+i}$

but $T\left( i \right)=0$ hence $T$ maps the upper half plane to the inside unit disk