A one-sided power sum inequality

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In this note we prove results of the following types. Let be given distinct complex numbers zⱼ satisfying the conditions |zⱼ| = 1, zⱼ = 1 for j=1,..., n and for every zⱼ there exists an i such that zᵢ = zⱼ. Then ₖ ∑ⱼ₌₁ⁿ zⱼᵏ ≤ - 1. If, moreover, none of the numbers zⱼ is a root of unity, then ₖ ∑ⱼ₌₁ⁿ zⱼᵏ ≤ - 2/π³ n. The constant -1 in the former result is the best possible. The above results are special cases of upper bounds for ₖ ∑ⱼ₌₁ⁿ bⱼzⱼᵏ obtained in this paper.