We are here concerned with the oscillatory behavior of solutions of the equation (1) x" + a(t)x = 0, where a(t) is locally integrable on [0, oo ). Our main result is an extension of a nonoscillation theorem due to Hartman [2, Theorem II], the contrapositive of which is a useful criterion for equation (1) Simple examples of averaging pairs are (1, 1), ((f+1)-1, t + 1), ((¿ + 1)~1/2, log (/ + 1)). In fact, any pair (