Prove Complex Inequality: $(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|)>=\sqrt{2}$

[siddjain]

Prove that $$(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|) >= \sqrt{2}$$

 

[Delta2]

Apply the inequality $$|A|+|B|\geq |A+B|$$ for $A=z_1+z_2, B=z_1-z_2$

Then apply it again for $A=z_1+z_2,B=z_2-z_1$.

You ll get two inequalities, add them and it should be straightforward to proceed.

EDIT: Well, using the above suggestion I think you can only prove a lower bound of 1 not $\sqrt{2}$.

 

[anuttarasammyak]

The given condition says
$$z_1=e^{i\phi_1}\cos\theta$$
$$z_2=e^{i\phi_2}\sin\theta$$
How about substituting them in the forlmula?