finding all the mobius transformation.?

Question

find all the mobius transformation that map the units disc$ D$ onto the left half plane

$H^-$ = {$w \in C : Re w <0 $}

My attempts : I know that all the Möbius transformations can be written on the form

$$w=f(z)=\frac{az+b}{cz+d}$$

here i don't know how can i find all the mobius transformation .

as im not getting in my heads pliz help me

Answer 1

Hint: It's a standard result that the Möbius transformations which map the unit disk onto itself are those of the form$$z\mapsto\omega\frac{z-a}{1-\overline az},$$with $|\omega|=1$ (there's a proof here). Now, find one Möbius which maps the left half plane onto the unit disk and you're don (well, almost).