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MUIRHEAD'S AND HOLLAND'S INEQUALITIES OF MIXED POWER MEANS FOR POSITIVE REAL NUMBERS

  • LEE, HOSOO (Department of Mathematics, College of Natural Sciences, Sungkyunkwan University) ;
  • KIM, SEJONG (Department of Mathematics, College of Natural Sciences, Chungbuk National University)
  • Received : 2016.08.25
  • Accepted : 2016.11.03
  • Published : 2017.01.30

Abstract

We review weighted power means of positive real numbers and see their properties including the convexity and concavity for weights. We study the mixed power means of positive real numbers related to majorization of weights, which gives us an extension of Muirhead's inequality. Furthermore, we generalize Holland's conjecture to the power means.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea (NRF)

References

  1. G.H. Hardy, J.E. Littlewood, G. Plya: Inequalities. Cambridge University Press, Cambridge 1934.
  2. F. Holland, On a mixed arithmetic-mean, geometric-mean inequality, Mathematics Competitions 5 (1992), 60-64.
  3. R. Horn and C. Johnson, Matrix Analysis, Cambridge University Press, 1985.
  4. H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), 509-541. https://doi.org/10.1002/cpa.3160300502
  5. K. Kedlaya, Proof of a mixed arithmetic-mean, geometric mean inequality, Amer. Math. Monthly 101 (1994) 355357.
  6. J. Lawson and Y. Lim, Karcher means and Karcher equations of positive definite operators, Trans. Amer. Math. Soc. Series B, Vol. 1 (2014), 1-22.
  7. Y. Lim and M. Palfia, Matrix power means and the Karcher mean, J. Funct. Anal. 262 (2012), no. 4, 1498-1514. https://doi.org/10.1016/j.jfa.2011.11.012
  8. A.W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Application, New York, Academic Press, 1979.
  9. R.F. Muirhead, Some methods applicable to identities and inequalities of symmetric algebraic functions of n letters, Proceedings of the Edinburgh Mathematical Society, 21 (1903), 144-157.
  10. J. Park and S. Kim, Remarks on convergence of inductive means, J. Appl. Math. & Informatics, 34 (2016), No. 3-4, 285-294. https://doi.org/10.14317/jami.2016.28