One-sided power sum and cosine inequalities

In this note we prove results of the following types. Let be given distinct complex numbers zjzj satisfying the conditions |zj|=1,zj≠1|zj|=1,zj≠1 for j=1,…,nj=1,…,n and for every zjzj there exists an ii such that zi=zj¯. Then infk∑j=1nzjk≤−1. If, moreover, none of the ratios zi/zjzi/zj with i≠ji≠j is a root of unity, then infk∑j=1nzjk≤−1π4logn. The constant −1 in the former result is the best possible. The above results are special cases of upper bounds for infk∑j=1nbjzjk obtained in this paper.