If `z_1 and z_2` are two complex numbers such that `|z_1|ltgt1lt|z_2|` then prove that `|(1-z_1barz_2)/(z_1-z_2)|lt1`

Explanation

Given `|z_1| lt 1 "and" |z_2|gt 1`
Then , to prove
`|(1-z_1barz_2)/(z_1-z_2)|lt 1" "["using"|(z_1)/(z_2)|=|z_1|/|z_2|]`
`rArr |1-z_1bar z_2|`
On squaring both sides we get ,
`(1-z_1bar z_2)(1-bar z_1 z_2) lt (z_1-z_2)(bar z_1 - bar z_2) [ "using" |z|^2=zbarz]`
`rArr 1-|z_1bar z_2-bar z_1 z_2+z_1barz_1z_2barz_2 lt z_1barz_1 - z_1barz_2-z_2bar z_2`
`rArr 1+|z_1|^2-|z_2|^2 lt |z_1|^2+|z_2|^2lt 0`
which is true by Eq. (i) as `|z_1| lt 1 and |z_2|gt 1`
`therefore (1-|z_1|^2)lt0 and (1-|z_2|^2)lt 0`
`therefore` Eq. (iii) is true whenever Eq. (ii) is true
`rArr |(1-z_1bar z_2)/(z_1-z_2)| lt 1`