Showing a complex inequality

Question

Let $z,w \in \mathbb C$ s.t $\bar z w \neq 1$ and $|z|\leq 1, |w| \leq 1$ then $ |\dfrac{z-w}{1-\bar z w}| \leq 1$

The hint is to show that $ |z-w|^2\leq |1-\bar z w|^2$ but i cant relate those 2

Answer 1

Use the fact that $|z|^2 = z \overline z$. Then,we have

$ |z - \overline \omega |^2 \le |1 - \overline z w|^2 \iff (z - \overline \omega) (\overline z - \omega) \le (1 - \overline z w) (1 - z \overline w) \iff z \overline z - z \overline \omega - \omega \overline \ + \omega \overline \omega \le 1 - z \overline \omega - \overline z \omega + z \overline z \omega \overline \omega $

After simplifying, the equivalence arrives at

$|z|^2 + |\omega|^2 <= 1 + |z|^2 |x|^2 \iff (|x| ^ 2 - 1) * (|y| ^ 2 - 1) \ge 0$, which is true since we assumed $|x| \le 1$ and $|y| \le 1$, so $|x| ^ 2 \le 1$ and $ |y| ^ 2 \le 1 $. So, we have proven the hint

Finally, since both terms are positive divide to get the conclusion squared. And since, the absolute value is always positive, you can get rid of the square, to get the desired conclusion.

Answer 2

$$|1-\overline zw|^2=(1-\overline zw)(1-\overline wz)=1+|wz|^2-\overline z w-\overline wz,$$ $$|z-w|^2=(z-w)(\overline z-\overline w)=|z|^2+|w|^2-z\overline w-w\overline z.$$ The difference is $$1+|wz|^2-|w|^2-|z|^2.$$