- Research Article
- Open Access
Schur-Convexity for a Class of Symmetric Functions and Its Applications
© W.-F. Xia and Y.-M. Chu. 2009
- Received: 16 May 2009
- Accepted: 14 September 2009
- Published: 12 October 2009
For , the symmetric function is defined by , where and are positive integers. In this article, the Schur convexity, Schur multiplicative convexity and Schur harmonic convexity of are discussed. As applications, some inequalities are established by use of the theory of majorization.
- Convex Function
- Symmetric Function
- Elementary Computation
- Complex Matrix
- Hermitian Matrix
Throughout this paper we use to denote the dimensional Euclidean space over the field of real numbers and . In particular, we use to denote
For the sake of convenience, we use the following notation system.
For , and , let
The notion of Schur convexity was first introduced by Schur in 1923 . It has many important applications in analytic inequalities [2–7], combinatorial optimization , isoperimetric problem for polytopes , linear regression , graphs and matrices , gamma and digamma functions , reliability and availability , and other related fields. The following definition for Schur convex or concave can be found in [1, 3, 7] and the references therein.
where denotes the th largest component in . is called Schur concave if is Schur convex.
However is called Schur multiplicatively concave if is Schur multiplicatively convex.
In paper , Anderson et al. discussed an attractive class of inequalities, which arise from the notion of harmonic convex functions. Here, we introduce the notion of Schur harmonic convexity.
for each pair of tuples and on , such that . is called a Schur harmonic concave function on if (1.6) is reversed.
The main purpose of this paper is to discuss the Schur convexity, Schur multiplicative convexity, and Schur harmonic convexity of the following symmetric function:
where , , and are positive integers. As applications, some inequalities are established by use of the theory of majorization.
In order to establish our main results we need several lemmas, which we present in this section.
The following lemma is so-called Schur's condition which is very useful for determining whether or not a given function is Schur convex or Schur concave.
for all and . Also is Schur concave if and only if (2.1) is reversed for all and . Here, is a symmetric function in meaning that for any and any permutation matrix .
for all . Also is Schur multiplicatively concave if and only if (2.3) is reversed.
for all . Also is Schur harmonic concave if and only if (2.4) is reversed.
From Definitions 1.1 and 1.3, we clearly see the fact that is Schur harmonic convex if and only if is Schur concave.
This fact, Lemma 2.1 and Remark 2.2 together with elementary calculation imply that Lemma 2.4 is true.
Lemma 2.6 (see ).
Lemma 2.7 (see ).
For the symmetric function is Schur concave in .
for all and
The proof is divided into four cases.
Therefore, (3.1) follows from Cases 1–4 and the proof of Theorem 3.1 is completed.
For the Schur multiplicative convexity or concavity of , we have the following theorem
It holds that is Schur multiplicatively concave in
for all and Then proof is divided into four cases.
Therefore, Theorem 3.2 follows from (3.10) and Cases 1–4.
From (3.11) and (3.12) we know that is Schur multiplicatively concave in and is Schur multiplicatively convex in .
For the symmetric function is Schur harmonic convex in .
for all and
The proof is divided into four cases.
Therefore, (3.15) follows from Cases 1–4 and the proof of Theorem 3.4 is completed.
In this section, we establish some inequalities by use of Theorems 3.1, 3.2 and 3.4 and the theory of majorization.
If , , , and , then
Therefore, Theorem 4.2 follows from (4.3) and Theorem 3.1 together with (1.7), and Theorem 4.2 follows from (4.3) and Theorem 3.4 together with (1.7).
If we take in Theorem 4.2 and in Theorem 4.2, respectively, then we have the following corollary.
Therefore, Theorem 4.5 follows from (4.7) and Theorem 3.4 together with (1.7), and Theorem 4.5 follows from (4.7) and Theorem 3.1 together with (1.7).
If we take and in Theorem 4.5, respectively, then we get the following corollary.
Therefore, Theorem 4.7 follows from (4.10), Theorem 3.2, and (1.7).
If we take and in Theorem 4.7, respectively, then we get the following corollary.
From Remark 3.3 and (4.10) together with (1.7) we clearly see that inequality in Corollary 4.8 is reversed for and inequality in Corollary 4.8 is true for .
Therefore, Theorem 4.11 follows from (4.15), Theorems 3.1, 3.4, and (1.7).
Mitrinovi et al. [21, pages 473–479] established a series of inequalities for and , . Obvious, our inequalities in Theorem 4.11 are different from theirs.
Therefore, Theorem 4.13 and follows from (4.17), Theorems 3.1, 3.4, and (1.7).
Therefore, Theorem 4.13 follows from (4.18), Theorem 3.2, and (1.7).
Therefore, Theorem 4.13 follows from (4.19), and Theorem 3.2 together with (1.7).
Therefore, Theorem 4.13 follows from (4.21), Theorem 3.2, and (1.7).
This work was supported by the National Science Foundation of China (no. 60850005) and the Natural Science Foundation of Zhejiang Province (no. D7080080, Y607128, Y7080185).
- Schur I: Über eine klasse von mittelbildungen mit anwendungen auf die determinantentheorie. Sitzungsberichte der Berliner Mathematischen Gesellschaft 1923, 22: 9–20.MATHGoogle Scholar
- Hardy GH, Littlewood JE, Pólya G: Some simple inequalities satisfied by convex functions. Messenger of Mathematics 1929, 58: 145–152.MATHGoogle Scholar
- Stepolhkepniak C: An effective characterization of Schur-convex functions with applications. Journal of Convex Analysis 2007,14(1):103–108.MathSciNetGoogle Scholar
- Aujla JS, Silva FC: Weak majorization inequalities and convex functions. Linear Algebra and Its Applications 2003, 369: 217–233.MathSciNetView ArticleMATHGoogle Scholar
- Guan K: The Hamy symmetric function and its generalization. Mathematical Inequalities & Applications 2006,9(4):797–805.MathSciNetView ArticleMATHGoogle Scholar
- Guan K-Z: Schur-convexity of the complete symmetric function. Mathematical Inequalities & Applications 2006,9(4):567–576.MathSciNetView ArticleMATHGoogle Scholar
- Guan K-Z: Some properties of a class of symmetric functions. Journal of Mathematical Analysis and Applications 2007,336(1):70–80. 10.1016/j.jmaa.2007.02.064MathSciNetView ArticleMATHGoogle Scholar
- Hwang FK, Rothblum UG: Partition-optimization with Schur convex sum objective functions. SIAM Journal on Discrete Mathematics 2004,18(3):512–524. 10.1137/S0895480198347167MathSciNetView ArticleMATHGoogle Scholar
- Zhang X-M: Schur-convex functions and isoperimetric inequalities. Proceedings of the American Mathematical Society 1998,126(2):461–470. 10.1090/S0002-9939-98-04151-3MathSciNetView ArticleMATHGoogle Scholar
- Stepolhkepniak C: Stochastic ordering and Schur-convex functions in comparison of linear experiments. Metrika 1989,36(5):291–298.MathSciNetView ArticleGoogle Scholar
- Constantine GM: Schur convex functions on the spectra of graphs. Discrete Mathematics 1983,45(2–3):181–188. 10.1016/0012-365X(83)90034-1MathSciNetView ArticleMATHGoogle Scholar
- Merkle M: Convexity, Schur-convexity and bounds for the gamma function involving the digamma function. The Rocky Mountain Journal of Mathematics 1998,28(3):1053–1066. 10.1216/rmjm/1181071755MathSciNetView ArticleMATHGoogle Scholar
- Hwang FK, Rothblum UG, Shepp L: Monotone optimal multipartitions using Schur convexity with respect to partial orders. SIAM Journal on Discrete Mathematics 1993,6(4):533–547. 10.1137/0406042MathSciNetView ArticleMATHGoogle Scholar
- Montel P: Sur les fonctions convexes et les fonctions sousharmoniques. Journal de Mathématiques 1928,7(9):29–60.MATHGoogle Scholar
- Niculescu CP: Convexity according to the geometric mean. Mathematical Inequalities & Applications 2000,3(2):155–167.MathSciNetView ArticleMATHGoogle Scholar
- Chu Y, Zhang X, Wang G: The Schur geometrical convexity of the extended mean values. Journal of Convex Analysis 2008,15(4):707–718.MathSciNetMATHGoogle Scholar
- Anderson GD, Vamanamurthy MK, Vuorinen M: Generalized convexity and inequalities. Journal of Mathematical Analysis and Applications 2007,335(2):1294–1308. 10.1016/j.jmaa.2007.02.016MathSciNetView ArticleMATHGoogle Scholar
- Marshall AW, Olkin I: Inequalities: Theory of Majorization and Its Applications, Mathematics in Science and Engineering. Volume 143. Academic Press, New York, NY, USA; 1979.MATHGoogle Scholar
- Wu S-H: Generalization and sharpness of the power means inequality and their applications. Journal of Mathematical Analysis and Applications 2005,312(2):637–652. 10.1016/j.jmaa.2005.03.050MathSciNetView ArticleMATHGoogle Scholar
- Bullen PS: A Dictionary of Inequalities. Volume 97. Longman, Harlow, UK; 1998.MATHGoogle Scholar
- Mitrinovic DS, Pecaric JE, Volenec V: Recent Advances in Geometric Inequalities. Volume 28. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1989.View ArticleMATHGoogle Scholar
- Weyl H: Inequalities between the two kinds of eigenvalues of a linear transformation. Proceedings of the National Academy of Sciences of the United States of America 1949, 35: 408–411. 10.1073/pnas.35.7.408MathSciNetView ArticleMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.