Question

Prove that $\left|\frac{z_1-z_2}{1-\overline{z_1}{z}_2}\right|<1$ if $\left|z_1\right|<1$ and $\left|z_2\right|<1$.

Answer 1

$\left|\frac{z_1-z_2}{1-{\overline{z}}_1{z}_2}\right|<1$
${|z_1-z_2|}^2<{|1-{\overline{z}}_1{z}_2|}^2$
$|z_1{|}^2+|z_2{|}^2-z_1\overline{z_2}-\overline{z_1}{z}_2<1-\overline{z_1}{z}_2-z_1\overline{z_2}+z_1\overline{z_1}{z}_2\overline{z_2}$

$\Rightarrow |z_1{|}^2+|z_2{|}^2<1+|z_1{|}^2|z_2{|}^2$

$\Rightarrow 1-|z_1{|}^2-|z_2{|}^2+|z_1{|}^2|z_2{|}^2>0$

$\Rightarrow (1-|z_1{|}^2)(1-|z_2{|}^2)>0$

Above equation can be true only if $|z_1|<1\text{and}|z_2|<1$. Hence proved.