On a preorder relation for Schur-convex functions and a majorization inequality for their gradients and divergences
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Aequationes Mathematicae
On a preorder relation for Schur-convex functions and a majorization inequality for their gradients and divergences Marek Niezgoda
Abstract. In this paper, a preorder relation for Schur-convex functions on Rn is introduced. A majorization statement is shown for the gradients and divergences of two Gateaux differentiable Schur-convex functions, provided that the difference of the involved functions is also Schur-convex. This implies the monotonicity of a related operator with respect to the used preorder and the classical majorization preorder on Rn . Special cases of the main result are also studied. In particular, applications are given for strongly convex functions. Some comparisons of variances are presented for uniform distribution. Mathematics Subject Classification. Primary 26B25, 26D10; Secondary 06F20 . Keywords. Majorization, Schur-convex function, Gradient, Divergence, Monotone operator, Convex function, Strongly convex function, Variance.
1. Majorization and Schur-convex functions In this introductory section, we collect some facts. Throughout the Euclidean space Rn is n inner product x, y = i=1 xi yi for x = (y1 , . . . , yn )∈Rn . A vector y = (y1 , . . . , yn ) ∈ Rn is said to (x1 , . . . , xn ) ∈ Rn (written as y ≺ x), if k i=1
y[i] ≤
k i=1
notation, definitions and basic equipped with the standard (x1 , . . . , xn ) ∈ Rn and y = be majorized by a vector x =
x[i] for all k = 1, . . . , n, and
n i=1
yi =
n
xi
i=1
(see [3, p. 8]). Here and after by z[1] , . . . , z[n] we mean the entries of a vector z = (z1 , . . . , zn ) ∈ Rn arranged in decreasing order, i.e., z[1] ≥ · · · ≥ z[n] . We also denote
AEM
M. Niezgoda
z↓ = (z[1] , . . . , z[n] ).
(1)
That is, z↓ is the rearrangement of z with its entries arranged in descending order. So, there exists an n-by-n permutation matrix p0 satisfying z = p0 z↓ . It is well-known (see [3, p. 10]) that for x, y ∈ Rn , y ≺ x iff y ∈ conv Pn x,
(2)
where the symbol Pn stands for the set of all n-by-n permutation matrices, and conv Pn x stands for the convex hull of the set Pn x = {px : p ∈ Pn }. The relation ≺ is Pn -invariant on Rn , i.e., for x, y ∈ Rn , y ≺ x iff p1 y ≺ p2 x for p1 , p2 ∈ Pn .
(3)
In particular, we have y ≺ x iff y↓ ≺ x↓ .
(4)
In the sequel, whenever C is a convex cone in a real linear space and a, b lie in this space, we write b ≤C a iff a − b ∈ C. We call ≤C the cone preorder induced by C. We introduce the sets D = Rn↓ = {x = (x1 , . . . , xn ) ∈ Rn : x1 ≥ · · · ≥ xn },
int D = {x = (x1 , . . . , xn ) ∈ Rn : x1 > · · · > xn }, dual D = {v ∈ Rn : v, d ≥ 0 for all d ∈ D}.
(5)
n
Notice that D is a closed convex cone in R , int D is the interior of D, and dual D is the dual cone of D. It is known (see [1,2]) that if x, y ∈ D, then y ≺ x iff y ≤dual D x iff x − y ∈ dual D.
(6)
n
In general, for any x, y ∈ R , y ≺ x iff x↓ ≤dual D y↓ iff x↓ − y↓ , d ≥ 0 for all d ∈ D. n
(7) n
A real function Φ : R → R is said to be Schur-convex if for x, y ∈ R , y ≺ x implies Φ(y) ≤ Φ(x). As can be seen in
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