On inequality $\mathrm{\mid }{z}^n-1\mathrm{\mid }\ge \mathrm{\mid }z-1\mathrm{\mid }$

It is known that inequality $\mathrm{\mid }{z}^n-1\mathrm{\mid }\ge \mathrm{\mid }z-1\mathrm{\mid }$ holds on the circle $|z-1/2|= 1/2$,where$n$isapositiveinteger\mathrm{.}Weprovethatinfact$n$ 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 $2\pi \mathrm{/}(n+1)\le x<\pi \mathrm{/}2$.