- Research Article
- Open Access

# The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications

- Yuming Chu
^{1}Email author and - Yupei Lv
^{1}

**2009**:838529

https://doi.org/10.1155/2009/838529

© Y. Chu and Y. Lv. 2009

**Received:**2 April 2009**Accepted:**20 May 2009**Published:**28 June 2009

## Abstract

We prove that the Hamy symmetric function is Schur harmonic convex for . As its applications, some analytic inequalities including the well-known Weierstrass inequalities are obtained.

## Keywords

- Convex Function
- Symmetric Function
- Concave Function
- Nonempty Interior
- Dimensional Euclidean Space

## 1. Introduction

Throughout this paper we use to denote the -dimensional Euclidean space over the field of real numbers, and .

Here and denote the classical arithmetic and geometric means, respectively.

discussed the Schur concavity Schur convexity of , and established some analytic inequalities.

The main purpose of this paper is to investigate the Schur harmonic convexity of the Hamy symmetric function . Some analytic inequalities including Weierstrass inequalities are established.

## 2. Definitions and Lemmas

Schur convexity was introduced by Schur in 1923 [6], and it has many important applications in analytic inequalities [7–12], linear regression [13], graphs and matrices [14], combinatorial optimization [15], information-theoretic topics [16], Gamma functions [17], stochastic orderings [18], reliability [19], and other related fields.

For convenience of readers, we recall some definitions as follows.

Definition 2.1.

A set is called a convex set if whenever . A set is called a harmonic convex set if whenever .

It is easy to see that is a harmonic convex set if and only if is a convex set.

Definition 2.2.

Let be a convex set a function is said to be convex on if for all . Moreover, is called a concave function if is a convex function.

Definition 2.3.

Let be a harmonic convex set a function is called a harmonic convex (or concave, resp.) function on if for all .

Definitions 2.2 and 2.3 have the following consequences.

Fact A.

is a harmonic concave function.

Definition 2.4.

where denotes the th largest component in . is called a Schur concave function on if is a Schur convex function on .

Definition 2.5.

for each pair of and in , such that .

Definitions 2.4 and 2.5 have the following consequences.

Fact B.

Let be a set, and , then is a Schur harmonic convex (or concave, resp.) function on if and only if is a Schur concave (or convex, resp.) function on .

The notion of generalized convex function was first introduced by Aczél in [20]. Later, many authors established inequalities by using harmonic convex function theory [21–28]. Recently, Anderson et al. [29] discussed an attractive class of inequalities, which arise from the notation of harmonic convex functions.

The following well-known result was proved by Marshall and Olkin [6].

Theorem 2 A.

for all and . Here, is a symmetric set means that implies for any permutation matrix .

Remark 2.6.

The following Lemma 2.7 can easily be derived from Fact, Theorem and Remark 2.6 together with elementary computation.

Lemma 2.7.

for all .

Next we introduce two lemmas, which are used in Sections 3 and 4.

Lemma 2.8 ([5, page 234]).

Lemma 2.9 ([2, Lemma 2.2]).

## 3. Main Result

In this section, we give and prove the main result of this paper.

Theorem 3.1.

The Hamy symmetric function is Schur harmonic convex in .

Proof.

To prove (3.1), we consider the following possible cases for .

Case 1 ( ).

Then (1.2) leads to , and (3.1) is clearly true.

Case 2 ( ).

and therefore, (3.1) follows from (3.2).

Case 3 ( ).

Simple computation yields

Therefore, (3.1) follows from (3.5) and the fact that is increasing in .

Case 4 ( ).

Differentiating (3.6) with respect to and , respectively, and using Lemma 2.8, we get

From (3.7) we obtain

Therefore, (3.1) follows from (3.8) and the fact that is increasing in .

## 4. Applications

In this section, making use of our main result, we give some inequalities.

Theorem 4.1.

Proof.

The proof follows from Theorem 3.1 and Lemma 2.9 together with (1.2).

If taking and in Theorem 4.1, respectively, then we have the following corollaries.

Corollary 4.2.

Corollary 4.3.

Taking in Corollaries 4.2 and 4.3, respectively, we get the following.

Corollary 4.4.

Corollary 4.5 (Weierstrass inequalities [30, Page 260]).

Theorem 4.6.

Proof.

Therefore, Theorem 4.6 follows from Theorem 3.1, (4.7) ,and (1.2).

Theorem 4.7.

Proof.

Therefore, Theorem 4.7 follows from Theorem 3.1, (4.10), and (1.2).

## Declarations

### Acknowlegments

This research is partly supported by N S Foundation of China under Grant 60850005 and N S Foundation of Zhejiang Province under Grants D7080080 and Y607128.

## Authors’ Affiliations

## References

- Hara T, Uchiyama M, Takahasi S-E:
**A refinement of various mean inequalities.***Journal of Inequalities and Applications*1998,**2**(4):387–395. 10.1155/S1025583498000253MathSciNetMATHGoogle Scholar - Guan K:
**The Hamy symmetric function and its generalization.***Mathematical Inequalities & Applications*2006,**9**(4):797–805.MathSciNetView ArticleMATHGoogle Scholar - Jiang W-D:
**Some properties of dual form of the Hamy's symmetric function.***Journal of Mathematical Inequalities*2007,**1**(1):117–125.MathSciNetView ArticleMATHGoogle Scholar - Ku H-T, Ku M-C, Zhang X-M:
**Inequalities for symmetric means, symmetric harmonic means, and their applications.***Bulletin of the Australian Mathematical Society*1997,**56**(3):409–420. 10.1017/S0004972700031191MathSciNetView ArticleMATHGoogle Scholar - Bullen PS:
*Handbook of Means and Their Inequalities, Mathematics and Its Applications*.*Volume 560*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:xxviii+537.View 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:xx+569.Google Scholar - Hardy GH, Littlewood JE, Pólya G:
**Some simple inequalities satisfied by convex functions.***Messenger of Mathematics*1929,**58:**145–152.MATHGoogle 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 - Aujla JS, Silva FC:
**Weak majorization inequalities and convex functions.***Linear Algebra and Its Applications*2003,**369:**217–233.MathSciNetView ArticleMATHGoogle Scholar - Qi F, Sándor J, Dragomir SS, Sofo A:
**Notes on the Schur-convexity of the extended mean values.***Taiwanese Journal of Mathematics*2005,**9**(3):411–420.MathSciNetMATHGoogle Scholar - Chu Y, Zhang X:
**Necessary and sufficient conditions such that extended mean values are Schur-convex or Schur-concave.***Journal of Mathematics of Kyoto University*2008,**48**(1):229–238.MathSciNetMATHGoogle 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 - Stepniak C:
**Stochastic ordering and Schur-convex functions in comparison of linear experiments.***Metrika*1989,**36**(5):291–298.MathSciNetView ArticleMATHGoogle 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 - 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 - Forcina A, Giovagnoli A:
**Homogeneity indices and Schur-convex functions.***Statistica*1982,**42**(4):529–542.MathSciNetMATHGoogle 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 - Shaked M, Shanthikumar JG, Tong YL:
**Parametric Schur convexity and arrangement monotonicity properties of partial sums.***Journal of Multivariate Analysis*1995,**53**(2):293–310. 10.1006/jmva.1995.1038MathSciNetView 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 - Aczél J:
**A generalization of the notion of convex functions.***Det Kongelige Norske Videnskabers Selskabs Forhandlinger, Trondheim*1947,**19**(24):87–90.MathSciNetMATHGoogle Scholar - Vamanamurthy MK, Vuorinen M:
**Inequalities for means.***Journal of Mathematical Analysis and Applications*1994,**183**(1):155–166. 10.1006/jmaa.1994.1137MathSciNetView ArticleMATHGoogle Scholar - Roberts AW, Varberg DE:
*Convex Functions, Pure and Applied Mathematics*.*Volume 5*. Academic Press, New York, NY, USA; 1973:xx+300.Google Scholar - Niculescu CP, Persson L-E:
*Convex Functions and Their Applications. A Contemporary Approach, CMS Books in Mathematics/Ouvrages de Mathématiques de la SMC, 23*. Springer, New York, NY, USA; 2006:xvi+255.Google Scholar - Matkowski J:
**Convex functions with respect to a mean and a characterization of quasi-arithmetic means.***Real Analysis Exchange*2004,**29**(1):229–246.MathSciNetMATHGoogle Scholar - Bullen PS, Mitrinović DS, Vasić PM:
*Means and Their Inequalities, Mathematics and Its Applications (East European Series)*.*Volume 31*. D. Reidel, Dordrecht, The Netherlands; 1988:xx+459.Google Scholar - Das C, Mishra S, Pradhan PK:
**On harmonic convexity (concavity) and application to non-linear programming problems.***Opsearch*2003,**40**(1):42–51.MathSciNetMATHGoogle Scholar - Das C, Roy KL, Jena KN:
**Harmonic convexity and application to optimization problems.***The Mathematics Education*2003,**37**(2):58–64.MathSciNetGoogle Scholar - Kar K, Nanda S:
**Harmonic convexity of composite functions.***Proceedings of the National Academy of Sciences, Section A*1992,**62**(1):77–81.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 - Bullen PS:
*A Dictionary of Inequalities, Pitman Monographs and Surveys in Pure and Applied Mathematics*.*Volume 97*. Longman, Harlow, UK; 1998:x+283.MATHGoogle Scholar

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