If `z_1a n dz_2` are two complex numbers and `c >0` , then prove that `|z_1+z_2|^2lt=(1+c)|z_1|^2+(1+c^(-1))|z_2|^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_1a n dz_2 are complex numbers and u=sqrt(z_1z_2) , then prove that |z_1|+|z_2|=|(z_1+z_2)/2+u|+|(z_1+z_2)/2-u|

For any two complex numbers z_1 and z_2 prove that: |\z_1+z_2|^2 +|\z_1-z_2|^2=2[|\z_1|^2+|\z_2|^2]

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

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

For any two complex numbers z_1 and z_2 prove that: |\z_1+z_2|^2=|\z_1|^2+|\z_2|^2+2Re bar 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 omega is the nth root of unity and Z_1 , Z_2 are any two complex numbers , then prove that . Sigma_(k=0)^(n-1)| z_1+ omega^k z_2|^2=n{|z_1|^2+|z_2|^2} where n in N

If z_(1), z_(2) in C (set of complex numbers), prove that |z_(1) + z_(2)| le |z_(1)| + |z_(2)|

For any two complex numbers z_1 and z_2 , prove that Re(z_1z_2)=Re(z_1) Re(z_2)-Im(z_1) Im(z_2) .