Bounds for Products of Zeros of Solutions to Nonhomogeneous ODE with Polynomial Coefficients

Abstract

We consider the equation $u^{″}=P(z)u+F(z)\hspace{0.17em}\hspace{0.17em}(z\in \mathbb{C})$, where $P(z)$ is a polynomial and $F(z)$ is an entire function. Let $z_k(u)\hspace{0.17em}\hspace{0.17em}(k=\mathrm{1,2},\dots )$ be the zeros of a solution $u(z)$ to that equation. Lower estimates for the products ${\prod }_{k=\mathrm{1}}^j|z_k(u)|\hspace{0.17em}\hspace{0.17em}(j=\mathrm{1,2},\dots )$ are derived. In particular, they give us a bound for the zero free domain. Applications of the obtained estimates to the counting function of the zeros of solutions are also discussed.