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On two integral inequalities*

Published online by Cambridge University Press:  14 February 2012

E. T. Copson
E. T. Copson
Affiliation:
University of St Andrews
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Synopsis

In 1932, Hardy and Littlewood [1] proved the inequality

The constant 4 is best possible; equality occurs when f(x) = A Y(Bx), where

y(x) = e−½x sin (x sin y−y) (y = ⅓π), (x ≧ o)

and A and B (>0) are constants. In [2], three proofs are given. The inequality has also been discussed in [3, 4]. A very elementary proof in which the function Y(x) emerges naturally is given in this paper.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 77 , Issue 3-4 , 1977 , pp. 325 - 328
Copyright
Copyright © Royal Society of Edinburgh 1977

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References

1Hardy, G. H. and Littlewood, J. E.. Some integral inequalities connected with the calculus of variations. Quart. Jl Math. Oxford Ser. 3 (1932), 241–252.CrossRefGoogle Scholar
2Hardy, G. H., Littlewood, J. E. and Pólya, G.. Inequalities (Edinburgh: Univ. Press, 1934).Google Scholar
3Kato, T.. On an inequality of Hardy, Littlewood and Pólya. Advances in Math. 7 (1971), 217–218.CrossRefGoogle Scholar
4Everitt, W. N.. On an extension to an integro-differential equation of Hardy, Littlewood and Pólya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1971), 295–333.Google Scholar