# Prove inequality for all complex numbers

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.

• It is also obvious for $|z|=1$. – ajotatxe Oct 23 '18 at 15:38

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 $$|z| = |z-z^2+z^2| \le |z-z^2|+|z^2|=|z||z-1|+|z^2|\le|z-1|+|z^2|$$ when $$|z|\le1$$.
Note also, $$|z^2|=|z|^2$$.