Prove that there exists no complex number z such that `|z| < 1/3 and sum_(n=1)^n a_r z^r =1`, where `| a_r|< 2`.

Prove that there exists no complex number z such that |z|<(1)/(3) and sum_(n=1)^(n)a_(r)z^(r)=1, where |a_(r)|<2

Prove that |(z-1)/(1-barz)|=1 where z is as complex number.

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

If z_(1) , z_(2) are two complex numbers such that |(z_(1)-z_(2))/(z_(1)+z_(2))|=1 and iz_(1)=Kz_(2) , where K in R , then the angle between z_(1)-z_(2) and z_(1)+z_(2) is

If z_1 and z_2 are two complex numbers such that |z_1|lt1lt|z_2| then prove that |(1-z_1barz_2)/(z_1-z_2)|lt1

If z_(1),z_(2),z_(3) are three nonzero complex numbers such that z_(3)=(1-lambda)z_(1)+lambda z_(2) where lambda in R-{0} then prove that points corresponding to z_(1),z_(2) and z_(3) are collinear.

If z_(1),z_(2),z_(3) are three non-zero complex numbers such that z_(3)=(1-lambda)z_(1)+lambda z_(2) where lambda in R-{0}, then points corresponding to z_(1),z_(2) and z_(3) is