Let #Z_1,Z_2# be complex numbers with #|Z_1| = |Z_2| = 1#, prove that #|Z_1 + 1| + |Z_2 +1| +|Z_1Z_2 +1| >=2#?

1 Answer
Feb 3, 2018

See below.

Explanation:

If #abs(Z_k) = 1 rArr Z_k = e^(i phi_k)#

then using de Moivre's identity

#e^(i phi) = cos phi + i sin phi#

#abs(Z_k+1) = sqrt((cos phi_k + 1)^2+sin^2 phi_k)# or simplifying

#(cos phi_k + 1)^2+sin^2 phi_k = 2(1+cos phi_k) = 4cos^2(phi_k/2)# so

#abs(Z_k+1) = 2 abs(cos(phi_k/2))# then

#|Z_1 + 1| + |Z_2 +1| +|Z_1Z_2 +1| = 2 (abs(cos(phi_1/2))+ abs(cos(phi_2/2))+abs(cos((phi_1+phi_2)/2)))#

but

# 1 le abs(cos(alpha))+abs(cos(beta))+ abs(cos(alpha+beta)) le 3#

then

#|Z_1 + 1| + |Z_2 +1| +|Z_1Z_2 +1| ge 2#