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`.

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) 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

Let z_1,z_2, z_3, z_n be the complex numbers such that |z_1|=|z_2|=|z_n|=1. If z=(sum_(k=1)^n z_k)(sum_(k=1)^n1/(z_k)) then proves that z is a real number

Let z_1z_2, z_3 z_n be the complex numbers such that |z_1|=|z_2||z_n|=1. If z=(sum_(k=1)^n z_k)(sum_(k=1)^n1/(z_k)) then proves that z is a real number

If z_1a n dz_2 are two complex numbers such that |z_1|=|z_2|a n d arg(z_1)+a r g(z_2)=pi , then show that z_1,=-( barz )_2dot

For any two complex number z_1a n d\ z_2 prove that: |z_1-z_2|lt=|z_1|+|z_2|

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 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