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Prove following inequality for all complex numbers:
\lvert z\rvert \le \lvert z \rvert ^2 + \lvert z-1 \rvert
It is obvious for $\lvert z\rvert \gt 1 $ but what about the rest ?
Any hints would be appreciated.
Hint: Write the inequality as $|z| (1 - |z|) \le |z-1|$. If $|z| \le 1$,
$|z| (1-|z|) \le 1 - |z|$.
Try the triangle inequality
|z| = |z-z^2+z^2| \le |z-z^2|+|z^2|=|z||z-1|+|z^2|\le|z-1|+|z^2|
Note also, $|z^2|=|z|^2$.
Required, but never shown