Sums of zeros of solutions to non-homogeneous ODE with polynomial coefficients

We consider the equation u″=P(z)u+F(z)u″=P(z)u+F(z) (z∈Cz∈C), where P(z)P(z) is a polynomial and F(z)F(z) is an entire function. Let zk(u)zk(u), k=1,2,…k=1,2,…, be the zeros of a solution u(z)u(z) to that equation. Estimates for the sums ∑k=1j1|zk(u)| (j=1,2,…j=1,2,…) are derived. In particular, they give us bounds for the zero free domain. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed.