NECESSARY AND SUFFICIENT CONDITIONS FOR THE SCHUR HARMONIC CONVEXITY OR CONCAVITY OF THE EXTENDED MEAN VALUES

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) is an element of (0, infinity) x (0, infinity) if and only if (r, s) is an element of {(r, s) : s >= 1,s >= r, s + r + 3 >= 0} boolean OR {(r, s) : r >= -1, r >= s, s + r + 3 >= 0} (or {(r, s) : s <= -1, r <= -1, s + r + 3 <= 0}, respectively).