Abstract
We proved that if (M, τ) is a semi-finite von Neumann algebra, and x₀, …, xₙ₋₁ are τ-measurable operators, f is a nonnegative function on [0, ∞), and α₀, …, αₙ₋₁ are positive real numbers such that ∑ αⱼ xⱼ = 0 and ∑ αⱼ = 1. Then, if (i) f(0) = 0 and g(t) = f(√t) is convex, we have for each ℓ ∈ {0, …, n−1}: f(√(α_ℓ / (1−α_ℓ)) · |x_ℓ|) + ∑ f(√(αⱼ αₖ / 2(1−α_ℓ)) · |xⱼ − xₖ|) ≼ ∑ αⱼ f(|xⱼ|), where S_ℓ = {0, …, n−1} \ {ℓ}. (ii) Similarly, in the concave case:∑ αⱼ f(|xⱼ|) ≼ f(√(α_ℓ / (1−α_ℓ)) · |x_ℓ|) + ∑ f(√(αⱼ αₖ / 2(1−α_ℓ)) · |xⱼ − xₖ|).