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. $\sqrt[3]{32}$
• C. $2\sqrt{2}$
• D. $1$

Answer 1

The correct option is A $2$

$|z^3+\frac{1}{z^3}|\le 2$
But we know that $|z^3+\frac{1}{z^3}|\le |z{|}^3+\frac{1}{|z{|}^3}$
So when $|z|=1$ by $AM\ge GM$
$|z{|}^3+\frac{1}{|z{|}^3}=2$
Now $|z+\frac{1}{z}|\le |z|+\frac{1}{|z|}=2$
Maximum value $=2$