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

Abstract

It is known that inequality $|z^n-1|\ge |z-1|$ 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: $co{s}^{n}x<1-sinx$ for real $n>3$ and $2\pi /(n+1)\le x<\pi /2$.