On inequality $|z^n-1|\geq|z-1|$

It is known that inequality $|z^n-1|\geq|z-1|$ holds on the circle $|z-1/2|= 1/2$, where $n$ is a positive integer. We prove that in fact $n$ can be real number not less then 1. We also prove following inequality as a lemma: $cos^nx\lt 1-sinx$ for real $n\gt3$ and $2\pi/(n+1)\leq x \lt \pi/2$.