Note on a Theorem of Nehari

hFih, x) > hFih, x) (0 è h < h < oo). In the course of this investigation the question arose whether a C2 solution of (1), with F subject to (2), is uniquely determined by the conditions (C) yia) = y'ib) = 0, y(x) > 0 for x E (a, b\. Nehari conjectured that this solution is unique under an additional condition on the behavior of Fit, x) as a function of x. The purpose of this note is to show that such is the case if Fit, x) satisfies the following condition: (2d) For each fixed positive p and 0 ^Xi 0. (b) y,(x) >0 and y{ (x) >0 for x in (a, c). Then yi(x) = y2(x) on a^xyi(x) for some x in (a, c). Then by the meanvalue theorem there exists an x2 in (a, x) such that y^fe) >y'i(Xi) and by a second application an x3 in (a, x2) such that y2'(x3)>yi'(x3). Because of (2b) and the form of (1), however, this implies y2(x3) ■ <yi(xz). Received by the editors April 6, 1961. 407