A question of complex numbers inequality $ \overline{z_1}^2 z_2 + z_1^2 \overline{z_2} \leq C ( |z_1|^3 + |z_2|^3) $ Can I find a positive constant $C$ such that $$ \overline{z_1}^2 z_2 + z_1^2 \overline{z_2} \leq C ( |z_1|^3 + |z_2|^3) $$for any complex numbers $z_1, z_2$? 
Here the overline denotes its complex conjugate.

Write $z_j:=r_je^{i\theta_j}$, $j\in\{1,2\}$. Then 
$$\bar z_1^2z_2+z_1^2\bar z_2\leqslant 2\max\{r_1,r_2\}^3|\cos(\theta_2-2\theta_1)|,$$
hence the claimed inequality is true (with the optimal constant $C=1$).
