Integral inequalities for second-order linear oscillation

We present several results related to the classical Lyapunov inequality for the oscillation of second-order linear equations. The first is an improved Lyapunov inequality given in terms of the downswing of the functions integral(a)(t)(t-a)q(t) dt and integral(t)(b)(b-t)q(t) dt , extending earlier results of Kwong and Harris and Kong. Nonoscillation criteria are derived as corollaries. A Lyapunov-type inequality for two consecutive zeros of the derivative of a solution is then established and a nonoscillation criterion given as a corollary. An oscillation criterion for positive q(t) is also proved. It extends the known condition integral t(gamma) q(t) dt = infinity , gamma is an element of (0, 1) .