Hostname: page-component-848d4c4894-8bljj Total loading time: 0 Render date: 2024-06-22T09:19:28.205Z Has data issue: false hasContentIssue false
  • English
  • Français

A general version of the Hardy–Littlewood–Polya–Everitt (HELP) inequality

Published online by Cambridge University Press:  14 November 2011

Christer Bennewitz
Christer Bennewitz
Affiliation:
Department of Mathematics, University of Uppsala, Uppsala, Sweden
Get access Rights & Permissions [Opens in a new window]

Synopsis

The inequality (0·1) below is naturally associated with the equation −(pu′)′ + qu = λu. By assuming that one end-point of the interval (a, b) is regular and the other limit-point for this equation, Everitt characterized the best constant K in tems of spectral properties of the equation. This paper sketches a theory for more general inequalities (0·2), (0·3) similarly related to the equation Su = λTu. Here S and T are ordinary, symmetric differential expressions. A characterization of the best constants in (0·2), (0·3) is given which generalises that of Everitt.

For the case when S is of order 1 and T is multiplication by a positive function, all possible inequalities are given together with the best constants and cases of equality. Furthermore, an example is given of a valid inequality (0·1) on an interval with both end-points regular for the corresponding differential equation. This contradicts a conjecture by Everitt and Evans. Finally, the general theory for the left-definite inequality (0·3) is specialised to the case when S is a Sturm-Liouville expression. A family of examples is given for which the best constants can be explicitly calculated.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 97 , 1984 , pp. 9 - 20
Copyright
Copyright © Royal Society of Edinburgh 1984

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Atkinson, F. V., Bennewitz, C., Everitt, W. N. and Race, D.. The Titchmarsh-Weyl m-coefficient: a survey of properties and extensions. To appear.Google Scholar
2Bennewitz, C.. Generalisations of some statements of Kodaira. Proc. Roy. Soc. Edinburgh Sect. A 71 (1972/1973), 113–119.Google Scholar
3Bennewitz, C.. A generalisation of Niessens's limit-circle criterion. Proc. Roy. Soc. Edinburgh Sect. A 78 (1977), 81–90.CrossRefGoogle Scholar
4Bennewitz, C.. Spectral theory for pairs of differential operators. Ark. Mat. 15 (1977), 33–61.Google Scholar
5Evans, W. D. and Everitt, W. N.. A return to the Hardy-Littlewood integral inequality. Proc. Roy. Soc. London. Ser. A 380 (1982), 447–486.Google Scholar
6Evans, W. D. and Zettl, A.. Norm inequalities involving derivatives. Proc. Roy. Soc. Edinburgh Sect. A 82 (1978), 51–70.Google Scholar
7Everitt, W. N.. On an extension to an integro-differential inequality of Hardy, Littlewood and Polya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1971/1972), 295–333.Google Scholar
8Everitt, W. N. and Zettl, A.. On a class of integral inequalities. J. London Math. Soc. 17 (1978), 291–303.CrossRefGoogle Scholar
9Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge University Press, 1934).Google Scholar
10Russell, A.. On a fourth order singular integral inequality. Proc. Roy. Soc. Edinburgh Sect. A 80 (1978), 249–260.Google Scholar