Solution of an open problem for Schur convexity or concavity of the Gini mean values

The Schur convexity or concavity problem of the Gini mean values S(a, b; x, y) with respect to (x, y) ∈ (0, ∞) × (0, ∞) for fixed (a, b) ∈ ℝ × ℝ is still open. In this paper, we prove that S(a, b; x, y) is Schur convex with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): a ⩾ 0, b ⩾ 0, a + b ⩾ 1}, and Schur concave with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (a, b) ∈ {(a, b): b ⩽ 0, b ⩽ a, a + b ⩽ 1} ∩ {(a, b): a ⩽ 0, a ⩽ b, a + b ⩽ 1}.