The schur convexity of gini mean values in the sense of harmonic mean

We prove that the Gini mean values S(a, b; x, y) are Schur harmonic convex with respect to (x,y) ∈ (0,∞) × (0,∞) if and only if (a,b) ∈ {(a,b) : a ≥ 0, a ≥ b,a+b+1 ≥ 0} ∪ {(a,b) : b ≥ a, a+b+1 ≥ 0} and Schur harmonic concave with respect to (x,y) ∈ (0,∞) × (0,∞) if and only if (a,b) ∈ {(a,b) : a ≤ 0, b ≤ 0,a+b+1 ≤ 0}.