Ukrainian Mathematical Journal

, Volume 71, Issue 2, pp 248–258 | Cite as

On the Cèsaro and Copson Norms of Nonnegative Sequences

  • V. I. KolyadaEmail author

The Cèsaro and Copson norms of a nonnegative sequence are the lp -norms of its arithmetic means and the corresponding conjugate means. It is well known that, for 1 < p < 1, these norms are equivalent. In 1996, G. Bennett posed the problem of finding the best constants in the associated inequalities. The solution of this problem requires the evaluation of four constants. Two of them were found by Bennett. We find one of the two unknown constants and also prove one optimal weighted-type estimate for the remaining constant.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Bennett, “Factorizing the classical inequalities,” Mem. Amer. Math. Soc., 120, No. 576 (1996).MathSciNetCrossRefGoogle Scholar
  2. 2.
    C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Boston (1988).zbMATHGoogle Scholar
  3. 3.
    S. Boza and J. Soria, “Solution to a conjecture on the norm of the Hardy operator minus the identity,” J. Funct. Anal., 260, 1020–1028 (2011).MathSciNetCrossRefGoogle Scholar
  4. 4.
    G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd edn., Cambridge Univ. Press, Cambridge (1967).zbMATHGoogle Scholar
  5. 5.
    V. I. Kolyada, “Optimal relationships between L p-norms for the Hardy operator and its dual,” Ann. Mat. Pura Appl. (4), 4, No. 2, 423–430 (2014).MathSciNetCrossRefGoogle Scholar
  6. 6.
    N. Kruglyak and E. Setterqvist, “Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions, Proc. Amer. Math. Soc., 136, 2005–2013 (2008).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Karlstad UniversityKarlstadSweden

Personalised recommendations