A one-sided power sum inequality

TL;DR

This paper establishes bounds on the infimum of sums of powers of complex numbers on the unit circle, revealing optimal constants and extending to weighted sums, with implications for complex analysis and number theory.

Contribution

It proves new bounds for sums of powers of complex numbers on the unit circle, including the best possible constant -1 and a logarithmic bound for non-root of unity cases.

Findings

Infimum of sum of powers is at most -1, best possible constant.

For non-root of unity complex numbers, the infimum is bounded by - (2/π^3) log n.

Results extend to weighted sums of complex numbers.

Abstract

In this note we prove results of the following types. Let be given distinct complex numbers $z_j$ satisfying the conditions $|z_j| = 1, z_j \not= 1$ for $j=1,..., n$ and for every $z_j$ there exists an $ i$ such that $z_i = \bar{z_j}. $ Then $$\inf_{k} \sum_{j=1}^n z_j^k \leq - 1. $$ If, moreover, none of the numbers $z_j$ is a root of unity, then $$\inf_{k} \sum_{j=1}^n z_j^k \leq - \frac {2} {\pi^3} \log n. $$ The constant -1 in the former result is the best possible. The above results are special cases of upper bounds for $\inf_{k} \sum_{j=1}^n b_jz_j^k$ obtained in this paper.