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_(3))| lt 1`

Explanation

Let us assume the result
i.e, `|(1-z_(1)barz_(2))/(z_(1) -z_(2))|lt 1`
` Rightarrow (|1-z_(1)barz_(2)|)/(|z_(1)-z_(2)|)lt1`
`Rightarrow |1-z_(1)barz_(2)|^(2)lt|z_(1) -z_(2)|^(2)`
`Rightarrow (1-z_(1)barz_(2))(1-barz_(1)z_(2))lt(z_(1)-z_(2))(barz_(1)-barz_(2))`
`Rightarrow 1-barz_(1)z_(2) -z_(1)barz_(2)+z_(1)barz_(2)+ z_(1)barz_(1)z_(2)barz_(2) lt z_(1)barz_(2)-z_(1)barz_(2) -z_(1)barz_(2)=z_(2)barz_(2)`
`1-|z_(1)|^(2)-|z_(2)|^(2)+|z_(1)|^(2)|z^(2)|^(2)lt0`
`(1- |z_(1)|^(2))(1-|z_(2)|^(2))lt 0`
`Rightarrow (|z_(1)|^(2) -1)(|z_(2)|^(2)-1)lt0`
`rArr |z_(1)|^(2) lt 1 lt |z_(2)|`
`rArr |z_(1)| lt 1 lt |z_(2)|` as |z| can't be negative which is he case.
or `|z_(2)| lt 1 lt |z_(1)|`