Question

If $\left|\frac{{(Z_1-Z_2)}^2}{{(1-{\overline{Z_1}Z}_2)}^2}\right|<1$ then prove that, $\left|Z_1\right|>1$ and $\left|Z_2\right|>1$ or $\left|Z_1\right|<1$ and $\left|Z_2\right|<1$.

Answer 1

$|Z1-Z2{|}^2<|1-\overline{Z}1Z2{|}^2$

$=(Z1-Z2)(\overline{Z}1-\overline{Z}2)<(1-Z1\overline{Z}2)(1-\overline{Z}1Z2)$

$=|Z1{|}^2-Z1\overline{Z}2-Z2\overline{Z}1+|Z2{|}^2<1-Z1\overline{Z}2-\overline{Z}1Z2+|Z1{|}^2|Z2{|}^2$

$=|Z1{|}^2+|Z2{|}^2-|Z1{|}^2|Z2{|}^2-1<0$

$=(|Z1{|}^2-1)(1-|Z2{|}^2)<0$

$CASE-\left(1\right)$

$|Z1{|}^2-1>0and1-|Z2{|}^2<0$

$CASE-\left(2\right)$

$|Z1{|}^2-1<0AND1-|Z2{|}^2>0$

$hereeither(|Z1|>1and|Z2|>1)or(|Z1|<1and|Z2|<1)$