# Minimum values of complex number $|z_1^3+z_2^3|$ if $|z_1+z_2|$ and $|z_1^2+z_2^2|$ are given

If $$z_1$$ and $$z_2$$ are two complex numbers such that $$|z_1+z_2|=1$$ and $$|z_1^2+z_2^2|=25$$, find the minimum value of $$|z_1^3+z_2^3|$$.

My try:

The minimum value is $$37$$ which I dot by taking $$z_1+z_2=1$$ and considering only real part and$$z_1^2+z_2^2=25$$, hence we get $$z_1=4$$ and $$z_2=-3$$, solving we get $$|z_1^3+z_2^3|=|64-27|=37$$

But this is not the appropriate way , it needs to be solved via triangle property or property of complex number, I am not able to solve it via property

• A little observation: $$|z_1^2 + z_2^2| = |(z_1+z_2)^2 - 2z_1z_2| \le |z_1 + z_2|^2 + 2|z_1z_2|$$ This implies $$|z_1z_2| \ge 12$$ Oct 24, 2021 at 4:38
• Also use $|z_1^3+z_2^3|$ = $|z_1+z_2||z_1^2+z_2^2 - z_1z_2|$ Oct 24, 2021 at 4:50
• Nice, @eyllanesc and vivid. Hence,$$|z_1^3+z_2^3| = |(z_1+z_2)^2 - 3z_1z_2|\ge 3|z_1z_2| - |z_1+z_2|^2\ge 36-1 = 35.$$That's maybe not the solution, but it's getting close. In addition,$$25 = |2z_1z_2 - (z_1+z_2)^2|\ge 2|z_1z_2| - 1,$$ thus $|z_1z_2|\le 13$, so that $|z_1^3+z_2^3|\le 38$. Oct 24, 2021 at 5:35
• @eyllanesc $|z_1z_2| \ne 12$ but $\ge 12$ Oct 24, 2021 at 8:37
• $$|z_1^3+z_2^3| = |z_1^2-z_1z_2 + z_2^2| = |\frac32(z_1^2+z_2^2)-\frac12(z_1+z_2)^2|\\ \ge \frac32|z_1^2+z_2^2|-\frac12|z_1+z_2|^2 = 37$$ Oct 24, 2021 at 10:08

Observe that \begin{align} |z_1^3+z_2^3| &= |z_1^2-z_1z_2 + z_2^2| \\ &= \left|\frac32(z_1^2+z_2^2)-\frac12(z_1+z_2)^2\right| \\ &\ge \frac32|z_1^2+z_2^2|-\frac12|z_1+z_2|^2 \\ &= 37 \end{align} Also, note that the equality holds for $$(z_1,z_2) = (4,-3)$$. So, indeed, the minimum value of the required expression is $$37$$.