Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-25T18:09:47.801Z Has data issue: false hasContentIssue false
  • English
  • Français

On certain fourth-order integral inequalities

Published online by Cambridge University Press:  14 November 2011

A. Russell
A. Russell
Affiliation:
Timex Corporation, Dundee, Scotland
Get access Rights & Permissions [Opens in a new window]

Synopsis

The integral inequalities with which this paper is concerned are

and

where K(μ) and K1(μ) are positive numbers which depend on μ.

It is shown that each of these inequalities is valid only when μ = 0. Results relating to the corresponding inequalities on the extended interval (−∞, ∞) are also given.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 3-4 , 1979 , pp. 205 - 211
Copyright
Copyright © Royal Society of Edinburgh 1979

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

1Beckenbach, E. F. and Bellman, R.. Inequalities (Berlin: Springer, 1961).CrossRefGoogle Scholar
2Bradley, J. S. and Everitt, W. N.. On the inequality ‖f″‖≤K‖f‖ ‖f (4‖) Quart. J. Math. Oxford Ser. 25 (1974), 241–52.CrossRefGoogle Scholar
3Brodlie, K. W. and Everitt, W. N.. On an inequality of Hardy and Littlewood. Proc. Roy. Soc. Edinburgh Sect. A 72 (1975), 179–186.CrossRefGoogle Scholar
4Everitt, W. N.. Some positive definite differential operators. J. London Math. Soc. 43 (1968), 465–173.CrossRefGoogle Scholar
5Everitt, W. N.. On an extension to an integro-differential inequality of Hardy, Littlewood and Polya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1972), 295–333.Google Scholar
6Halperin, I.. Closures and adjoints of linear differential operators. Ann. Math. 38 (1937), 880–919.CrossRefGoogle Scholar
7Hardy, G. H. and Littlewood, J. E.. Some integral inequalities connected with the calculus of variations. Quart. J. Math. Oxford Ser. 3 (1932), 241–252.CrossRefGoogle Scholar
8Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge Univ Press, 1934).Google Scholar
9Ljubic, Ju. I.. On inequalities between the powers of a linear operator. Amer. Math. Soc. Transi 40 (1964), 39–84.Google Scholar
10Russell, A.. On a fourth-order singular integral inequality. Proc. Roy. Soc. Edinburgh Sect. A 80 (1978) 249–260.CrossRefGoogle Scholar
11Russell, A.. Integral inequalities and spectral theory of certain ordinary differential operators. (Dundee Univ. Ph.D. Thesis, 1975).Google Scholar