Open Access

On the constant in Meńshov-Rademacher inequality

Journal of Inequalities and Applications20062006:68969

DOI: 10.1155/JIA/2006/68969

Received: 26 March 2005

Accepted: 7 September 2005

Published: 25 April 2006

Abstract

The goal of the paper is twofold: (1) to show that the exact value in the Meńshov-Rademacher inequality equals 4/3, and (2) to give a new proof of the Meńshov-Rademacher inequality by use of a recurrence relation. The latter gives the asymptotic estimate .

[12345678910]

Authors’ Affiliations

(1)
Muskhelishvili Institute of Computational Mathematics, Georgian Academy of Sciences
(2)
Department of Statistics & Probability, Michigan State University

References

  1. Chobanyan S: Some remarks on the Men'shov-Rademacher functional. Matematicheskie Zametki 1996,59(5):787–790. translation in Mathematical Notes $59$ (1996), no. 5–6, 571–574 translation in Mathematical Notes $59$ (1996), no. 5-6, 571–574MathSciNetView ArticleGoogle Scholar
  2. Doob JL: Stochastic Processes. John Wiley & Sons, New York; 1953:viii+654.MATHGoogle Scholar
  3. Gohberg IC, Kreĭn MG: Theory and Applications of Volterra Operators in Hilbert Space. Izdat. "Nauka", Moscow; 1967:508. translated in Translations of Mathematical Monographs, vol.~24, American Mathematical Society, Province, RI, 1970 translated in Translations of Mathematical Monographs, vol.~24, American Mathematical Society, Province, RI, 1970Google Scholar
  4. Kounias EG: A note on Rademacher's inequality. Acta Mathematica Academiae Scientiarum Hungaricae 1970,21(3–4):447–448. 10.1007/BF01894790MATHMathSciNetView ArticleGoogle Scholar
  5. Kwapień S, Pełczyński A: The main triangle projection in matrix spaces and its applications. Studia Mathematica 1970, 34: 43–68.MATHMathSciNetGoogle Scholar
  6. Loève M: Probability Theory, The University Series in Higher Mathematics. 2nd edition. D. Van Nostrand, New Jersey; 1960:xvi+685.Google Scholar
  7. Meńshov D: Sur les séries de fonctions orthogonales, I. Fundamenta Mathematicae 1923, 4: 82–105.Google Scholar
  8. Móricz F, Tandori K: An improved Menshov-Rademacher theorem. Proceedings of the American Mathematical Society 1996,124(3):877–885. 10.1090/S0002-9939-96-03151-6MATHMathSciNetView ArticleGoogle Scholar
  9. Rademacher H: Einige Sätze über Reihen von allgemeinen Orthogonalfunktionen. Mathematische Annalen 1922,87(1–2):112–138. 10.1007/BF01458040MATHMathSciNetView ArticleGoogle Scholar
  10. Somogyi Á: Maximal inequalities for not necessarily orthogonal random variables and some applications. Analysis Mathematica 1977,3(2):131–139. 10.1007/BF01908425MATHMathSciNetView ArticleGoogle Scholar

Copyright

© Sergei Chobanyan et al. 2006

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.