<s>[INST] Prove this inequality in complex domain (5) Let $z_{1},z_{2},z_{3}\in C$ 
show that
$$|z_{1}+z_{2}+z_{3}|^2+|(z_{1}-z_{2})(z_{1}-z_{3})|+|(z_{2}-z_{3})(z_{2}-z_{1})|
+|(z_{3}-z_{1})(z_{3}-z_{2})|\le 3(|z_{1}|^2+|z_{2}|^2+|z_{3}|^2)$$
Iif prove following inequality,Use Cauchy-Schwarz inequality we have
$$(|z_{1}+z_{2}+z_{3}|^2\le (|z_{1}|+|z_{2}|+|z_{3}|)^2\le 3(|z_{1}|^2+|z_{2}|^2+|z_{3}|^2)$$
 [/INST] Let $O$, $A$, $B$ and $C$ be points in the plain and $M$ is a gravity canter of $\Delta ABC$.
Hence, 
$3(OA^2+OB^2+OC^2)=9OM^2+AB^2+AC^2+BC^2\geq$
$\geq9OM^2+AB\cdot AC+AB\cdot BC+AC\cdot BC$ and we are done!
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