Prove that function transforms $S$ into $S$

Question

I have to prove, that function $\Phi(z):=\frac{1-\overline a z}{z-a}$ transforms $S$ into $S$ where $S:=\{|z|=1\}$ and $|a|>1$

I don't know where can I start. Any hint?

Answer 1

Factor out the $z$ from the numerator, so you have $$z(\dfrac{1}{z} - \bar{a})$$

in the numerator. Now, what happens when you use the hint I gave you in the comment? Can you figure it out from there?

Answer 2

If $|z|=1$ then $$\Phi(z)=\frac{z(\overline z - \overline a)}{z-a}=\frac{z \overline{(z - a)}}{z-a}$$

And now $|\cdot|$ of the right hand side is equal to 1. Thanks!