Weird complex number inequality

Question

Let z1,z2 be complex numbers with $|z_1|=|z_2|=1$. Prove that $|z_1+1|+|z_2+1|+|z_1z_2+1| \ge 2$.

My attempt: I started with $z_1=e^{i\theta_1}$ and $z_2=e^{i\theta_2}$ and then plugged them into the equations given but as it came to $\theta_1+\theta_2$ in the last modulus I got pretty confused regarding how to proceed.

Please help me out.

Answer 1

Since $1=|\bar z_2| $, mutliplying with it we get $|z_1z_2+1|=|z_1+\bar z_2|=|-z_1-\bar z_2|$, while $|z_2+1|=|\bar z_2+1|$, so now the triangle inequality:

$|z_1+1|+|\bar z_2+1|+|-z_1-\bar z_2| \ge |1+1|=2$ does it!