Prove that this mobius transformation maps unit disc to unit disc

Question

This transformation question was asked in an assignment question and I was unable to solve it.

Prove that the transformation f(z) =$\frac { z-z_{0}} { \bar z_{0} z -1} $ maps unit disc to itself.

Attempt : I tried to assume that x+iy lies in U and then prove that f(z) also lies in U. But that's a very lenghty and primitive approach.

Can you please tell any better approach for this problem.

Thanks!!

Answer 1

Let $|z|=1$ be the unit cfircle. The transformation is $z=\frac{w-a}{\bar aw-1}$ Then $$|z|=1 \implies \left|\frac{w-a}{\bar a w-1} \right|=1 \implies |w-a|=|\bar a w-1|\implies (w-a)(\bar w-\bar a)=(\bar a w-1)(a \bar w-1). $$ $$\implies (a\bar a-1)(w\bar w-1)=0 \implies |w|=1, ~\text{if}~ |a|\ne 1.$$ So $z$ points move in a unit circle so will be $w$ points, provided $|a| \ne 1.$