Let y(t) be a nontrivial solution of the second order differential inequality y(t){(r(t)y'(t))' + f(t,y(t))} less than or equal to 0. We show that the zeros of y(t) are simple; y(t) and y'(t) have at most finite number of zeros on any compact interval [a,b] under suitable conditions on r and f. Using the main result, we establish some nonlinear maximum principles and a nonlinear Levin's comparison theorem, which extend some results of Protter, Weinberger, and Levin.
