On an integral inequality

Abstract
The general integral inequality with which this paper is concerned is [J ∞ a {p(x)f'(x) 2 +q(x)f(x)2}dx] 2 ∞ a f(x) 2 dxJ ∞ a {(p(x)f'(x))'-q(x)f(x)} 2 dx 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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