Skip to main content

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

Abstract

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.

[1234567891011121314151617181920]

References

  1. 1.

    Stolarsky KB: Generalizations of the logarithmic mean. Mathematics Magazine 1975, 48: 87–92. 10.2307/2689825

    MathSciNet  Article  MATH  Google Scholar 

  2. 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. 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. 4.

    Pittenger AO: The logarithmic mean invariables. The American Mathematical Monthly 1985,92(2):99–104. 10.2307/2322637

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Pearce CEM, Pečarić J, Šimić V: On weighted generalized logarithmic means. Houston Journal of Mathematics 1998,24(3):459–465.

    MathSciNet  MATH  Google Scholar 

  6. 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. 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. 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. 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. 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/1021477248

    MathSciNet  Article  MATH  Google Scholar 

  11. 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/1181069663

    MathSciNet  Article  MATH  Google Scholar 

  12. 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. 13.

    Kuang J-C: Applied Inequalities. 3rd edition. Shandong Science and Technology Press, Jinan, China; 2004.

    Google Scholar 

  14. 14.

    Mitrinović DS: Analytic Inequalities, Die Grundlehren der mathematischen Wisenschaften. Volume 1965. Springer, New York, NY, USA; 1970:xii+400.

    Google Scholar 

  15. 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. 16.

    Wang B-Y: Foundations of Majorization Inequalities. Beijing Normal University Press, Beijing, China; 1990.

    Google Scholar 

  17. 17.

    Zhang X-M: Geometrically Convex Functions. An'hui University Press, Hefei, China; 2004.

    Google Scholar 

  18. 18.

    Zhang X-M: Optimization of Schur-convex functions. Mathematical Inequalities & Applications 1998,1(3):319–330.

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Guan K: Schur-convexity of the complete elementary symmetric function. Journal of Inequalities and Applications 2006, 2006: 9 pages.

    Article  Google Scholar 

  20. 20.

    Hardy GH, Littlewood JE, Pólya G: Inequalities. 2nd edition. Cambridge University Press, Cambridge, UK; 1952:xii+324.

    Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ning-Guo Zheng.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Zheng, NG., Zhang, ZH. & Zhang, XM. Schur-Convexity of Two Types of One-Parameter Mean Values in Variables. J Inequal Appl 2007, 078175 (2007). https://doi.org/10.1155/2007/78175

Download citation

Keywords

  • Symmetric Function
  • Elementary Symmetric Function
\