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On an Integral Inequality

W. N. Everitt and D. S. Jones
Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences
Vol. 357, No. 1690 (Nov. 4, 1977), pp. 271-288
Published by: Royal Society
Stable URL: http://www.jstor.org/stable/79467
Page Count: 18
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On an Integral Inequality
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Abstract

The general integral inequality with which this paper is concerned is [∫a ∞ {p(x)f′(x)2 + q(x)f(x)2}dx]2 ≤ K(p,q) ∫a ∞ f(x)2 dx ∫a ∞ {(p(x)f′(x))′ - q(x)f(x)}2dx where the coefficients p and q are real-valued, with p positive, p′ continuous, q continuous and bounded below, on the half-line [a, ∞ ). Here K(p,q) is a positive number or +∞ and depends on the coefficients p and q. The general theory of this inequality shows that the best possible constant K(p,q) lies between the bounds 4 ≤ K(p,q) ≤ ∞ . One of the problems left unsolved in the general theory was whether or not all values of K between the bounds 4 and ∞ can be realized by making a suitable choice of the coefficients p and q. It is the object of this paper to show that an affirmative answer can be given to this problem; all values between 4 and ∞ can be realized.

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