If $z_1,z_2,z_3$ are complex numbers such that $\left|z_1\right|=\left|z_2\right|=\left|z_3\right|=1$. Then ${\left|z_1-z_2\right|}^2+{\left|z_2-z_3\right|}^2+{\left|z_3-z_1\right|}^2$ cannot be exceed _____

Let  $y={\left|z_1-z_2\right|}^2+{\left|z_2-z_3\right|}^2+{\left|z_3-z_1\right|}^2$

$=\left(z_1-z_2\right)\left({\overline{z}}_1-{\overline{z}}_2\right)+\left(z_2-z_3\right)\left({\overline{z}}_2-{\overline{z}}_3\right)+\left({\overline{z}}_3-{\overline{z}}_1\right)\left(z_3-z_1\right)$

$=6-\left(z_1{\overline{z}}_2+z_2{\overline{z}}_1+{\overline{z}}_2{z}_3+z_3{\overline{z}}_2+z_1{\overline{z}}_3+z_3{\overline{z}}_1\right)$

We know that

$\left|z_1+z_2+z_3\right|\ge 0$

$z_1{\overline{z}}_2+z_2{\overline{z}}_1+z_1{\overline{z}}_3+{\overline{z}}_1{z}_3+z_2{\overline{z}}_3+z_1{\overline{z}}_3\ge -3$

From 1 & 2

$y\le 9$