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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

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