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

TL;DR

This paper extends a known inequality involving complex numbers to real exponents greater than or equal to 1 and proves an auxiliary inequality relating cosine and sine functions for certain ranges of n and x.

Contribution

It generalizes the inequality |z^n - 1| ≥ |z - 1| from integer n to real n ≥ 1 and introduces a new inequality involving cosine and sine functions.

Findings

Inequality |z^n - 1| ≥ |z - 1| holds for real n ≥ 1.

Proved a lemma: cos^n x < 1 - sin x for n > 3 and specific x ranges.

Extended the applicability of a classical inequality to real exponents.

Abstract

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 $2\pi/(n+1)\leq x \lt \pi/2$.