Advertisement

Archiv der Mathematik

, Volume 99, Issue 4, pp 393–399 | Cite as

Sharp bounds for the difference between the arithmetic and geometric means

  • J. M. Aldaz
Article

Abstract

We present sharp bounds for \({\sum_{i=1}^n \alpha_i x_i -\prod_{i=1}^n x_i^{\alpha_i} }\) in terms of the variance of the vector \({(x_1^{1/2},\dots,x_n^{1/2})}\).

Mathematics Subject Classification (2000)

26D15 

Keywords

Variance Arithmetic-geometric inequality 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aldaz J.M.: Self-improvement of the inequality between arithmetic and geometric means, Journal of Mathematical Inequalities 3, 213–216 (2009) arXiv:0807.1788MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aldaz J.M.: A refinement of the inequality between arithmetic and geometric means, Journal of Mathematical Inequalities 2, 473–477 (2008) arXiv:0811.3145MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Aldaz J.M.: Concentration of the ratio between the geometric and arithmetic means, Journal of Theoretical Probability 23, 498–508 (2010) arXiv:0807.483MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Aldaz J.M.: A stability version of Hölder’s inequality, Journal of Mathematical Analysis and Applications 343, 842–852 (2008) arXiv:0710.2307MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Aldaz J.M.: Comparison of differences between arithmetic and geometric means, Tamkang J. of Math. 42, 453–462 (2011) arXiv:1001.5055.MathSciNetMATHGoogle Scholar
  6. 6.
    Aldaz J.M.: A measure-theoretic version of the Dragomir-Jensen inequality, Proc. Amer. Math. Soc. 140, 2391–2399 (2012) arXiv:1101.0239MathSciNetCrossRefGoogle Scholar
  7. 7.
    Alzer H.: A new refinement of the arithmetic mean-geometric mean inequality, Rocky Mountain J. Math. 27, 663–667 (1997)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cartwright D.I., Field M.J.: A refinement of the arithmetic mean-geometric mean inequality, Proc. Amer. Math. Soc. 71, 36–38 (1978)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Dragomir S.: Bounds for the normalised Jensen functional, Bull. Austral. Math. Soc. 74, 471–478 (2006)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Furuichi S.: On refined Young inequalities and reverse inequalities, J. Math. Inequal. 5, 21–31 (2011)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Mercer A.McD.: Bounds for the A-G, A-H, G-H, and a family of inequalities of Ky Fan’s type, using a general method, Journal of Mathematical Analysis and Applications 243, 163–173 (2000)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Departamento de MatemáticasUniversidad Autónoma de MadridMadridSpain

Personalised recommendations