Skip to main content


  • Research Article
  • Open Access

Schur-Convexity of Two Types of One-Parameter Mean Values in Variables

Journal of Inequalities and Applications20072007:078175

  • Received: 10 July 2007
  • Accepted: 9 November 2007
  • Published:


We establish Schur-convexities of two types of one-parameter mean values in variables. As applications, Schur-convexities of some well-known functions involving the complete elementary symmetric functions are obtained.


  • Symmetric Function
  • Elementary Symmetric Function


Authors’ Affiliations

Huzhou Broadcast and TV University, Huzhou, Zhejiang, 313000, China
Zixing Educational Research Section, Chenzhou, Hunan, 423400, China
Haining College, Zhejiang Broadcast and TV University, Haining, Zhejiang, 314400, China


  1. Stolarsky KB: Generalizations of the logarithmic mean. Mathematics Magazine 1975, 48: 87–92. 10.2307/2689825MathSciNetView ArticleMATHGoogle Scholar
  2. Alzer H: Über eine einparametrige familie von mittelwerten [On a one-parameter family of means] . Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte 1987, 1–9.Google Scholar
  3. Alzer H: Über eine einparametrige Familie von Mittelwerten. II. [On a one-parameter family of mean values. II]. Bayerische Akademie der Wissenschaften. Mathematisch-Naturwissenschaftliche Klasse. Sitzungsberichte 1988, 23–39.Google Scholar
  4. Pittenger AO: The logarithmic mean invariables. The American Mathematical Monthly 1985,92(2):99–104. 10.2307/2322637MathSciNetView ArticleMATHGoogle Scholar
  5. Pearce CEM, Pečarić J, Šimić V: On weighted generalized logarithmic means. Houston Journal of Mathematics 1998,24(3):459–465.MathSciNetMATHGoogle Scholar
  6. Xiao Z-G, Zhang Z-H: The Stolarsky mean ofpositive numbers. Journal of YueYang Normal University 2001,14(4):5–8.Google Scholar
  7. Xiao Z-G, Zhang Z-H, Qi F: A type of mean values of several positive numbers with two parameters. RGMIA Research Report Collection 2006.,9(2, article 11):Google Scholar
  8. Xiao Z-G, Zhang Z-H: The Heron mean ofpositive numbers. Journal of YueYang Normal University 2001,14(2):1–5.Google Scholar
  9. Xiao Z-G, Zhang Z-H, Lokesha V: The weighted Heron mean of several positive numbers. RGMIA Research Report Collection 2005.,8(3, article 6):Google Scholar
  10. Elezović N, Pečarić J: A note on Schur-convex functions. The Rocky Mountain Journal of Mathematics 2000,30(3):853–856. 10.1216/rmjm/1021477248MathSciNetView ArticleMATHGoogle Scholar
  11. Qi F: A note on Schur-convexity of extended mean values. The Rocky Mountain Journal of Mathematics 2005,35(5):1787–1793. 10.1216/rmjm/1181069663MathSciNetView ArticleMATHGoogle Scholar
  12. 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
  13. Kuang J-C: Applied Inequalities. 3rd edition. Shandong Science and Technology Press, Jinan, China; 2004.Google Scholar
  14. Mitrinović DS: Analytic Inequalities, Die Grundlehren der mathematischen Wisenschaften. Volume 1965. Springer, New York, NY, USA; 1970:xii+400.Google Scholar
  15. Roberts AW, Varberg DE: Convex Functions. Pure and Applied Mathematics. Volume 57. Academic Press, New York, NY, USA; 1973:xx+300.Google Scholar
  16. Wang B-Y: Foundations of Majorization Inequalities. Beijing Normal University Press, Beijing, China; 1990.Google Scholar
  17. Zhang X-M: Geometrically Convex Functions. An'hui University Press, Hefei, China; 2004.Google Scholar
  18. Zhang X-M: Optimization of Schur-convex functions. Mathematical Inequalities & Applications 1998,1(3):319–330.MathSciNetView ArticleMATHGoogle Scholar
  19. Guan K: Schur-convexity of the complete elementary symmetric function. Journal of Inequalities and Applications 2006, 2006: 9 pages.View ArticleGoogle Scholar
  20. Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1952:xii+324.MATHGoogle Scholar


© Ning-Guo Zheng et al. 2007

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.