Need help with proof with absolute value and complex numbers.

Question

Had some trouble trying to prove the following problem.

Prove that if $|z| < 1$ and $|w| < 1$, then

$$ \frac{|z-w|}{|1-\overline{z}w|} < 1 $$

Would appreciate some help.

Answer 1

It is equivalent to $\lvert z-w\rvert^2 <\lvert 1-\bar zw\rvert^2$, i.e.:

\begin{align*} &(z-w)(\bar z-\bar w) < (1-\bar zw)(1- z\bar w)\iff z\bar z+w\bar w <1+z\bar zw\bar w\\ \iff &1 -z\bar z-w\bar w+z\bar zw\bar w=(1-z\bar z)(1-w\bar w)=(1-\lvert z\rvert^2)(1-\lvert w\rvert^2)>0. \end{align*}