Resumen de Necessary and sufficient conditions for the Schur harmonic convexity or concavity of the extended mean values

Wei Xia, Yu-Ming Chu, Di Wang

• In this paper, we prove that the extended values E(r, s; x, y) are Schur harmonic convex (or concave, respectively) with respect to (x, y) ∈ (0, ∞) × (0, ∞) if and only if (r, s) ∈ {(r, s) : s ≥ −1, s ≥ r, s + r + 3 ≥ 0} ∪ {(r, s) : r ≥ −1, r ≥ s, s+r + 3 ≥ 0} (or {(r, s) : s ≤ −1, r ≤ −1, s+r + 3 ≤ 0}, respectively).