Research Article | Open | Published:
On Schur Convexity of Some Symmetric Functions
Journal of Inequalities and Applicationsvolume 2010, Article number: 543250 (2010)
For and , the symmetric function is defined as , where are positive integers. In this paper, the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of are discussed. As consequences, several inequalities are established by use of the theory of majorization.
Throughout this paper, we use the following notation system.
For , and , let
If then we denote by
Next we introduce some definitions and well-known results.
Let be a set, and a real-valued function on is called a Schur convex function if
for each pair of -tuples and in with , that is,
where denotes the th largest component of . A function is called Schur concave if is Schur convex.
Let be a set, and a function is called a Schur multiplicatively convex function on if
for each pair of -tuples and in with . is called Schur multiplicatively concave if is Schur multiplicatively convex.
Let be a set. A function is called a Schur harmonic convex (or Schur harmonic concave, resp.) function on if
for each pair of -tuples and in with .
Schur convexity was introduced by Schur  in 1923 and it has many applications in analytic inequalities [2, 3], extended mean values [4, 5], graphs and matrices , and other related fields. Recently, the Schur multiplicative convexity was investigated in [7–9] and the Schur hamonic convexity was discussed in .
For and , the symmetric function is defined as
where are positive integers.
The aim of this article is to discuss the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of the symmetric function .
Lemma 1.4 (see ).
Let be a continuous symmetric function. If is differentiable in , then is Schur convex in if and only if
Lemma 1.5 (see ).
Let be a continuous symmetric function. If is differentiable in , then is Schur multiplicatively convex in if and only if
Lemma 1.6 (see ).
Let be a continuous symmetric function. If is differentiable in , then is Schur harmonic convex in if and only if
Lemma 1.7 (see ).
Let and . If , then
Lemma 1.8 (see ).
Let and . If , then
Lemma 1.9 (see ).
Suppose that and If then
2. Main Result
The symmetric function is Schur convex, Schur multiplicatively convex, and Schur harmonic convex in for all
According to Lemmas 1.4–1.6 we only need to prove that
for all and
We divided the proof into seven cases.
If and , then (1.7) leads to
If and , then (1.7) yields
If , , and , then from (1.7) we clearly see that
If , , and , then (1.7) implies that
If , and , then (1.7) leads to
If , , and , then (1.7) yields
If , , and , then (1.7) implies
Therefore, inequality (2.1) follows from inequalities (2.5), (2.9), (2.13), (2.17), (2.21), (2.25), and (2.29), inequality (2.2) follows from inequalities (2.6), (2.10), (2.14),(2.18), (2.22), (2.26), and (2.30), and inequality (2.3) follows from inequalities (2.7), (2.11), (2.15), (2.19), (2.23), (2.27), and (2.31).
In this section, we establish several inequalities by use of Theorem 2.1 and the theory of majorization.
It follows from Lemmas 1.7, 1.8, 1.9, and Theorem 2.1 that Theorem 3.1 is obvious.
If , , and , then
If , , , and , then
Theorem 3.2 follows from Theorem 2.1 and the fact that
If we take and in Theorem 3.1(3) and Theorem 3.2, respectively, then we get the following.
If with and then
If we take and in Theorem 3.1(3) and Theorem 3.2, respectively, then one gets the following.
If with and then
If we take in Corollaries 3.3 and 3.4, then we have
If , then
Theorem 3.6 follows from Theorem 2.1 and (1.7) together with the fact that
If we take in Theorem 3.6, then we have the following.
If , then
Let be an -dimensional simplex in and let be an arbitrary point in the interior of . If is the intersection point of straight line and hyperplane , then
It is easy to see that and , and these identities imply that
Therefore, Theorem 3.8 follows from Theorem 2.1 and (1.7) together with (3.11).
Suppose that is a complex matrix, and are the eigenvalues of . If is a positive definite Hermitian matrix, then
We clearly see that and
Therefore, Theorem 3.9(1) follows from (1.7), (3.13), and the Schur convexity of , Theorems 3.9(2) and (3) follow from (3.14) and (3.15) together with the Schur multiplitively convexity of , and Theorem 3.9(4) follows from (3.16) and the Schur harmonic convexity of .
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The research was supported by NSF of China (no. 60850005) and the NSF of Zhejiang Province (nos. D7080080, Y7080185, and Y607128).