Question

If $n$ is a positive integer, then prove that $\left|Im\right(z^n\left)\right|\le n\left|Im\right(z\left)\right|{\left|z\right|}^{n-1}$.

Answer 1

Let $z=r.e^{i\theta }$
Hence $z^n=r^n.e^{in\theta }$
Hence $Im\left(z^n\right)$ $=r^n.sin\left(n\theta \right)$
Now, $n|Im\left(z\right)|.|z^{n-1}|$ $=n|rsin\theta |.|z^n|.\frac{1}{|z|}$
$=n{r}^n|sin\theta |$
Now
$n|sin\theta |\ge |sin\left(n\theta \right)|$ for $n\ge 1$
Hence the above inequality is true.