I am given two complex numbers $z$ and $w$ that satisfy the following constraint $$ |z - \overline{z}w| + |w|^2 < 1. $$
I want to see if the following inequality is true $$ z^2 \overline{w} + \overline{z}^2w + |w|^2(z^2 \overline{w} + \overline{z}^2w - 4|z|^2) \geq 0. $$ Is it possible for Mathematica to prove or disprove the above inequality?