Skip to main content

HELP integral and series inequalities

  • Chapter
General Inequalities 6

Abstract

A survey of recent work on the Hardy Everitt Littlewood and Pólya (HELP) inequality and analogous series inequalities investigated by Brown and Evans is presented. Included is a description of the numerical techniques devised by Brown, Kirby and Pryce to determine the best constants in the inequalities.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N I Akhiezer, The Classical Moment Problem. Oliver and Boyd, Edinburgh and London, 1965. (Translated from the Russian edition of 1960.)

    Google Scholar 

  2. F.V. Atkinson, Discrete and Continuous Boundary Problems. Academic Press, New York, 1964.

    Google Scholar 

  3. F.V. Atkinson, On the location of the Weyl circles. Proc. Roy. Soc. Edinb. A 88 (1981), 345–356.

    Google Scholar 

  4. P.B. Baily, M.K. Gordon and L.F. Shampine, Automatic solution of the Sturm-Liouville problem. ACM Trans Math. Software 4 (1978), 193–207.

    Article  Google Scholar 

  5. C. Bennewitz, Spectral theory for pairs of differential operators. Arkiv. för mat 15 (1977), 33–61.

    Google Scholar 

  6. C. Bennewitz, A general version of the Hardy-Littewood -Pólya-Everitt (HELP) inequality. Proc. Roy. Soc. Edinb (A) 97 (1984), 9–20.

    Article  Google Scholar 

  7. C. Bennewitz, The HELP inequality in the regular case. General Inequalities 5 (Proc. Oberwolfach, 1986; edited by W. Walter; Birkhäuser-Verlag, Basel, 1987 ).

    Google Scholar 

  8. B.M. Brown and W.D. Evans, On an extension of Copson’s inequality for infinite series. Preprint.

    Google Scholar 

  9. B.M. Brown, W.D. Evans and L.L. Littlejohn, Series inequalitites, orthogonal polynomials and the spectral theory of difference operators. Preprint.

    Google Scholar 

  10. B.M. Brown, V.G. Kirby, and J.D. Pryce, Numerical determination of the Titchmarsh-Weyl m-coefficient and its applciations to HELP inequalities. Proc. Roy. Soc. Lond. (A) 426 (1989), 167–188.

    Article  Google Scholar 

  11. B.M. Brown, V.G. Kirby, and J.D. Pryce, A numerical method for the determination of the Titchmarsh-Weyl m-coefficient. To appear in Proc. Roy. Soc. Lond.

    Google Scholar 

  12. E.T. Copson, Two series inequalities. Proc. Roy. Soc. Edinb. (A) 83 (1979), 109–114.

    Google Scholar 

  13. R. England, Error estimates for Runga-Kutta-type solutions to systems of ordinary differential equations. Compt. J. 12 (1969), 166–170.

    Article  Google Scholar 

  14. W.D. Evans and W.N. Everitt, A return to the Hardy-Littlewood inequality. Proc. Roy. Soc. Lond. (A) 380 (1982), 447–486.

    Article  Google Scholar 

  15. W.D. Evans, W.N. Everitt, W.K. Hayman and S. Ruscheweyh, On a class of integral inequalities of Hardy-Littlewood type. J. d’Anal. Math. 46 (1986), 118–147.

    Article  Google Scholar 

  16. W.D. Evans and W.N. Everitt, Hardy-Littlewood integral inequalities. Lecture Notes in Pure and Applied Mathematics 129 (1991), 29–51 (Marcel Dekker, Inc., New York, 1991; edited by W.N. Everitt).

    Google Scholar 

  17. W.D. Evans and W.N. Everitt, HELP inequalities for limit-circle and regular problems. Proc. Roy. Soc. Lond. A 432 (1991), 367–390.

    Article  Google Scholar 

  18. W.D. Evans and A. Zettl, Norm inequalities involving derivatives. Proc. Roy. Soc. Edinb. (A) 82 (1978), 51–70.

    Article  Google Scholar 

  19. W.N. Everitt, On an extension to an integro-differential inequality of Hardy, Littlewood and Pölya. Proc. Roy. Soc. Edinb. A 69 (1971/72), 295–333.

    Google Scholar 

  20. G.H. Hardy and J.E. Littlewood, Some inequalities connected with the calculus of variations. Quart. J. Math. (Oxford) (2) 3 (1932), 241–252.

    Article  Google Scholar 

  21. G.H. Hardy, J.E. Littlewood and G. Pólya, luequalitites. Cambridge Univ Press, 1934.

    Google Scholar 

  22. E. Hellinger, Zur Stieltjesschen Kettenbruchtheorie. Math. Ann. 86 (1922), 18–29.

    Google Scholar 

  23. D.B.Hinton and R.T.Lewis, Spectral analysis of second-order difference equations. J. Math. Anal. and Appl. 63 (1978), 421–438.

    Google Scholar 

  24. V.G. Kirby, A numerical method for determining the Titchmarsh-Weyl m-coefficient, and its applications to certain integro-differential inequalities. Ph.D. Thesis, University of Wales College of Cardiff, 1990.

    Google Scholar 

  25. H. Kuki, Complex gamma functions with error control. Comm ACM 15 (1972), 262–267.

    Google Scholar 

  26. R. Nevanlinna, Asymptotische Entwickelungen beschränkter Funktionen and das Stieltjessche Momentenproblem. Ann. Acad. Sci. Fenn. A 18 (5) (1922), 1–52.

    Google Scholar 

  27. H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen. Math. Ann. 68 (1910), 220–269.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1992 Springer Basel AG

About this chapter

Cite this chapter

Brown, B.M., Evans, W.D., Everitt, W.N. (1992). HELP integral and series inequalities. In: Walter, W. (eds) General Inequalities 6. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale d’Analyse Numérique, vol 103. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7565-3_23

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7565-3_23

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-7567-7

  • Online ISBN: 978-3-0348-7565-3

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics