New proofs of Schur-concavity for a class of symmetric functions

Definition 2. [1,2] Let x = (x1, ..., xn) and y = (y1, ..., yn) Î R . (i) x is said to be majorized by y (in symbols x ≺ y) if ∑k i=1 x[i] ≤ ∑k i=1 y[i] for k = 1, 2,..., n 1 and ∑n i=1 xi = ∑n i=1 yi , where x [1] ≥ · · · ≥ x[n] and y[1] ≥ · · · ≥ y[n] are rearrangements of x and y in a descending order. (ii) Let Ω ⊂ R, : Ω ® R is said to be a Schur-convex function on Ω if x ≺ y on Ω implies (x) ≤ (y). is said to be a Schur-concave function on Ω if and only if is Schur-convex function on Ω.