In this paper, we improve the Sturm comparison theorem and two nonoscillation criteria of LEIGHTON and WINTNER, and establish two variants of a WINTNER's nonoscillatory criterion of the second order linear differential equation [r(t)u'(t)]'+c(t)u(t) = 0, where r, c: [t(0),infinity) --> IR, r > 0 a.e. on [t(0), infinity) and 1/r, c is an element of L(1)(t(0), b) for each b is an element of (t(0), infinity) for some t(0) greater than or equal to 0. Using these two criteria, we improve some nonoscillation criteria of HARTMAN,; HILLE, MOORE, POTTER. WINTNER, and WILLETT. These proofs are more elegant and concise than those of theirs.
