If `z_(1) ` and `z_(2)` are two complex numbers such that `|z_(1)| lt 1 lt |z_(2)|`, then prove that `|(1- z_(1)barz_(2))//(z_(1)-z_(2))| lt 1`

Explanation

Given that ` |z_1| lt 1 lt |z_2 |`
Now, ` |(1- z_1barz_2 )/(z_ 1 - z _2 ) | lt 1 `
or ` |1 - z_1 barz_2 | lt |z_1 - z_2 |`
` or " " |1 - z_1 bar z_2|^(2) lt | z _1 - z_2|^(2)`
` or " " (1 - z_1barz_2 )(bar(1 - z_1 barz_2 ) lt (z_1 - z_2 )(bar(z_1 - z_2 )) `
or ` (1 - z_1 barz_2 ) (1 - barz_1 z_2) lt (z_ 1 - z_2 ) (bar z_1 - bar z _2 ) `
or ` " " 1 - z_1 barz_2 - bar z_1 z_2 + z_1 bar z _1 z_2 bar z _2 lt z _1 bar z_1 - z_1 bar z_2 - barz_1 z_2 + z_2 bar z _2 `
or ` 1 + |z_1 |^(2)|z_2|^(2) lt |z_1|^(2) + |z_2|^(2) `
or ` (1 - |z_1|^(2)) (1 - |z_2|^(2)) lt 0 `
which is obviously true is
`|z_1| lt 1 lt | z _2 | `
` rArr |z_1 | ^(2) lt 1 lt |z_2 | ^(2)`
` rArr (1 - |z_1|^(2)) gt 0 and ( 1 - |z_2|^(2)) lt 0 `