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XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya*

Published online by Cambridge University Press:  14 February 2012

W. N. Everitt
W. N. Everitt
Affiliation:
Department of Mathematics, The University, Dundee.
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Synopsis

This paper considers an extension of the following inequality given in the book Inequalities by Hardy, Littlewood and Polya; let f be real-valued, twice differentiable on [0, ∞) and such that f and f are both in the space fn, ∞), then f′ is in L,2(0, ∞) and

The extension consists in replacing f′ by M[f] where

choosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the form

where K is a positive number independent of f.

The analysis involves a fourth-order differential equation and the second-order equation associated with M.

A number of examples are discussed to illustrate the theorems obtained and to show that the extended inequality (*) may or may not hold.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 69 , Issue 4 , 1972 , pp. 295 - 333
Copyright
Copyright © Royal Society of Edinburgh 1972

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References

References to Literature

[1]Beckenbach, E. F. and Bellman, R., 1961. Inequalities. Berlin: Springer-Verlag.CrossRefGoogle Scholar
[2]Chaudhuri, Jyoti and Everitt, W. N., 1968. ‘On the spectrum of ordinary second-order differential operators’, Proc. Roy. Soc. Edinb., A, 68, 95–119.Google Scholar
[3]Chaudhuri, Jyoti and Everitt, W. N., 1969. ‘On the square of a formally self-adjoint differential expression’, J. Lond. Math. Soc., 1, 661–673.CrossRefGoogle Scholar
[4]Everitt, W. N., 1966. ‘On the limit-point classification of second-order differential operators’, J. Lond. Math. Soc, 41, 531–534.CrossRefGoogle Scholar
[5]Everitt, W. N. ‘On a property of the m-coefficient of a second-order linear differential equation’, J. Lond. Math. Soc. (to appear).Google Scholar
[6]Hardy, G. H., Littlewood, J. E. and Polya, G., 1934. Inequalities. Cambridge University Press.Google Scholar
[7]Hille, Einar, 1969. Lectures on ordinary differential equations. New York and London: Addison-Wesley.Google Scholar
[8]Naimark, M. A., 1968. Linear differential operators, Pt. II, New York: Ungar.Google Scholar
[9]Titchmarsh, E. C., 1962. Eigenfunction expansions associated with second-order differential equations, Pt. I, 2nd Edn. Oxford University Press.CrossRefGoogle Scholar
[10]Tranter, C. J., 1968. Bessel functions with some physical applications. London: English Universities Press.Google Scholar
[11]Weyl, Hermann, 1910. ‘Ueber gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlichen Funktionen’, Math. Annin, 68, 220–269.CrossRefGoogle Scholar
[12]Littlewood, J. E., 1947. Lectures on the theory of functions. Oxford University Press.Google Scholar
[13] (Additional reference added on the suggestion of a referee.) Kallman, R. R. and Rota, G-C., 1970. ‘On the inequality ’ in Inequalities II (Ed. Shisha, Oved), 187–192. New York and London: Academic Press.Google Scholar