complex no.

We have |z_1+z_2| = 4 \le |z_1|+|z_2|

Adding this to the two other similar inequalities we get |z_1|+|z_2|+|z_3| \ge 6

Now from Holder's inequality we obtain

(1+1+1)(1+1+1)(|z_1|^3+|z_2|^3+|z_3|^3) \ge (|z_1|+|z_2|+|z_3|)^3 \ge 6^3 = 216

|z_1|^3+|z_2|^3+|z_3|^3 \ge 24

But we are given that |z_1|^3+|z_2|^3+|z_3|^3 = 24

This means that |z_1|=|z_2|=|z_3| =2

and further since |z_1+z_2|=|z_1|+|z_2|, this means z₁ and z₂ are in the same line segment through origin

But since |z_1|=|z_2|, this also means that z_1=z_2

Using the second condition we get z_1=z_2=z_3=2 as the unique solution

Thanks bhatt sir and Shubodip

Thanks again sir!