A new numerical inequality for average power means is presented. Let α, β ∈ [-∞, +∞] and let a = (a,_k)k≥1 be a sequence of positive numbers. Consider the operator M_α(a) = (a₁^α+a₂^α + ⋯ + a_k^α/k)^1/α_k≥1. We denote by M_β o M_α the superposition of these operators. The following assertion is proved: if α < β, then M_β o M_α(a) ≤ M_α o M _β(a).