<s>[INST] How to relate the $|z^3+z^{-3}|$ with $|z+z^{-1}|$? I got stuck on this seemingly simple question:

If $z$ is a complex number satisfying $|z^3+z^{-3}| \le 2$, then the maximum possible value of $|z+z^{-1}|$ is:
  (A) $2$
  (B) $2^{1/3}$
  (C) $2\sqrt 2$
  (D) $1$

Using the AM-GM inequality, I showed that $|z|^3 + |z|^{-3} \ge 2$ and so I got:
$$ |z^3+z^{-3}| \le 2 \le |z|^3 + |z|^{-3}$$
I don't inderstand how I can remove the cubic power, and bring any of this in terms of $|z|$ and $|z|^{-1}$. What am I missing?
Thank you.
 [/INST] $(z+z^{-1})^3=z^3 + 3z + 3z^{-1}+z^{-3}$ therefore $|z+z^{-1}|^3 \le |z^3 + z^{-3}| + 3 |z+z^{-1}| \le 2 + 3|z+z^{-1}|$ and using notation $|z+z^{-1}| = x$ we have:
$x^3 \le 2 + 3x$ equivalent $(x-2)(x+1)^2 \le 0$ therefore $|z+z^{-1}| \le 2 $ is the answer
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