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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.

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  • $\begingroup$ HINT: Let $$Z_1=r(\cos A+i\sin A),z_2=R(\cos B+i\sin B)$$ $$\overline{z_1}^2 z_2 + z_1^2 \overline{z_2}$$ $$=r^2(\cos2A-i\sin2A)\cdot R(\cos B+i\sin B)+r^2(\cos2A+i\sin2A)\cdot R(\cos B-i\sin B) $$ $$=r^2\{\cos(-2A)+i\sin(-2A)\}\cdot R(\cos B+i\sin B)+r^2(\cos2A+i\sin2A)\cdot R\{\cos(-B)+i\sin(-B)\} $$ $$=r^2R\{\cos(B-2A)+i\sin(B-2A)+\cos(2A-B)+i\sin(2A-B)\}$$ $$=2r^2R\cos(2A-B)\text{ as }\cos(-x)=\cos x,\sin(-x)=-\sin x$$ $\endgroup$ – lab bhattacharjee Jun 24 '13 at 16:29
  • $\begingroup$ I actually removed the tag [complex-analysis], at it concerns mores complex numbers. $\endgroup$ – Davide Giraudo Jun 24 '13 at 16:50
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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 $C=1$).

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