Submajorization inequalities for τ-measurable operators involving convex and concave functions II

We prove the following submajorization inequalities for τ-measurable operators in a semifinite von Neumann algebra (M, τ).Let x₀, …, xₙ₋₁ be τ-measurable operators,f : [0, ∞) → [0, ∞) be a nonnegative function, and let α₀, …, αₙ₋₁ be positive real numbers such that (∑ⱼ₌₀ⁿ⁻¹ αⱼ xⱼ = 0), (∑ⱼ₌₀ⁿ⁻¹ αⱼ = 1), and αℓ ≥ ½ for some ℓ ∈ {0, …, n−1}. Then: (i) If f(0) = 0 and g(t) = f(√t) is convex, then ∑j,k ∈ S_ℓ [ f( √( αⱼ αk / (4 αℓ (1 − αℓ)) ) · |xⱼ + xk| ) +f( √( αⱼ αk (2 αℓ − 1) / (4 αℓ (1 − αℓ)) ) · |xⱼ − xk| ) ] ≼ ∑ⱼ₌₀ⁿ⁻¹ αⱼ f(|xⱼ|), where Sℓ = { 0, …, n−1 } ∖ { ℓ }. (ii) If h(t) = f(√t) is concave, then the reverse inequality holds: ∑ⱼ₌₀ⁿ⁻¹ αⱼ f(|xⱼ|) ≼ ∑j,k ∈ S_ℓ [ f( √( αⱼ αk / (4 αℓ (1 − αℓ)) ) · |xⱼ + xk| ) +f( √( αⱼ αk (2 αℓ − 1) / (4 αℓ (1 − αℓ)) ) · |xⱼ − xk| ) ], where Sℓ = { 0, …, n−1 } ∖ { ℓ }.