JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 97, 277-290 (1983) Generalizations of an Inequality of Kiguradze URI ELIAS Department of Mathematics, Technion-Israel Institute of Technology, Haifa 32000, Israel Submitted by R. P. Boas 1. INTRODUCTION A very useful, though simple, result about two term ordinary differential equations is the following proposition of Kiguradze Ill]: (a) Given the equation $+yF(x,y)=O (1.1) where F(x, y) has a fixed sign, positive or negative, for 0 <x < 00, --oo < y < 00. If y is a nonoscillatory solution of (1.1) on 10, 00 ), there exists an integer k, 0 < k < n such that y”’ > 0, i = O,..., k, (-ly’-ky’j’ > 0, j = k + l,..., n, (1.2) on a certain ray (a, co). Moreover, the parity of k is such that (-l)“pkF(x,y)<O. (b) ?f y”’ > 0, y’k+ ‘) < 0, i = O,..., k, (1.3) on (a, co) then (x - a) y”+ ‘) < (k - t) y(l), t = O,..., k, (1.4) on (a, co). Proposition (a) is widely used in the study of linear, nonlinear, and delay 277 0022-247x/83 $3.00 Copyright G 1983 by Acadermc Press. Inc. All rights of reproduction in any form reserved.