XXIII.—On an Extension to an Integro-differential Inequality of Hardy, Littlewood and Polya
- 1 January 1972
- journal article
- research article
- Published by Cambridge University Press (CUP) in Proceedings of the Royal Society of Edinburgh. Section A. Mathematical and Physical Sciences
- Vol. 69 (4) , 295-333
- https://doi.org/10.1017/s0080454100008797
Abstract
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, ∞) andThe extension consists in replacing f′ by M[f] wherechoosing f so that f and M[f] are in L2(0, ∞) and then seeking to determine if there is an inequality of the formwhere 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.Keywords
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