Sums of zeros of solutions to second order differential equations with polynomial coefficients

We consider the equation u\u27\u27=P(z)u, where P(z) is a polynomial. Let zk(u), k=1, 2, ... be the zeros of a solution u(z) to that equation. Inequalities for the sums ∑k=1j |zk(u)|-1 (j=1, 2, ...) are derived. They considerably improve the previous result of the author. Some applications of the obtained bounds are also discussed. An illustrative example is presented. It shows that the suggested results are sharp