- Research Article
- Open Access
On Schur Convexity of Some Symmetric Functions
© W.-F. Xia and Y.-M. Chu 2010
- Received: 20 November 2009
- Accepted: 3 March 2010
- Published: 8 March 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.
- Convex Function
- Intersection Point
- Arbitrary Point
- Symmetric Function
- Related Field
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.
where denotes the th largest component of . A function is called Schur concave if is Schur convex.
for each pair of -tuples and in with . is called Schur multiplicatively concave if is Schur multiplicatively convex.
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 ).
Lemma 1.5 (see ).
Lemma 1.6 (see ).
Lemma 1.7 (see ).
Lemma 1.8 (see ).
Lemma 1.9 (see ).
The symmetric function is Schur convex, Schur multiplicatively convex, and Schur harmonic convex in for all
for all and
We divided the proof into seven cases.
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 we take and in Theorem 3.1(3) and Theorem 3.2, respectively, then we get the following.
If we take and in Theorem 3.1(3) and Theorem 3.2, respectively, then one gets the following.
If we take in Theorem 3.6, then we have the following.
Therefore, Theorem 3.8 follows from Theorem 2.1 and (1.7) together with (3.11).
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 .
The research was supported by NSF of China (no. 60850005) and the NSF of Zhejiang Province (nos. D7080080, Y7080185, and Y607128).
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