- Research Article
- Open Access
The Schur Harmonic Convexity of the Hamy Symmetric Function and Its Applications
© Y. Chu and Y. Lv. 2009
- Received: 2 April 2009
- Accepted: 20 May 2009
- Published: 28 June 2009
- Convex Function
- Symmetric Function
- Concave Function
- Nonempty Interior
- Dimensional Euclidean Space
Schur convexity was introduced by Schur in 1923 , and it has many important applications in analytic inequalities [7–12], linear regression , graphs and matrices , combinatorial optimization , information-theoretic topics , Gamma functions , stochastic orderings , reliability , and other related fields.
For convenience of readers, we recall some definitions as follows.
Definitions 2.2 and 2.3 have the following consequences.
is a harmonic concave function.
Definitions 2.4 and 2.5 have the following consequences.
The notion of generalized convex function was first introduced by Aczél in . Later, many authors established inequalities by using harmonic convex function theory [21–28]. Recently, Anderson et al.  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 .
Theorem 2 A.
The following Lemma 2.7 can easily be derived from Fact, Theorem and Remark 2.6 together with elementary computation.
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]).
In this section, we give and prove the main result of this paper.
and therefore, (3.1) follows from (3.2).
Simple computation yields
From (3.7) we obtain
In this section, making use of our main result, we give some inequalities.
The proof follows from Theorem 3.1 and Lemma 2.9 together with (1.2).
Corollary 4.5 (Weierstrass inequalities [30, Page 260]).
Therefore, Theorem 4.6 follows from Theorem 3.1, (4.7) ,and (1.2).
Therefore, Theorem 4.7 follows from Theorem 3.1, (4.10), and (1.2).
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.
- 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
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.