Bounds for products of zeros of solutions to nonhomogeneous ODE with polynomial coefficients (Q274778)

Summary: We consider the equation \[ u'' = P(z) u + F(z)\,(z \in \mathbb{C}), \] where \(P(z)\) is a polynomial and \(F(z)\) is an entire function. Let \(z_k(u)(k = 1,2, \ldots)\) be the zeros of a solution \(u(z)\) to that equation. Lower estimates for the products \(\mathbf\Pi_{k = 1}^j | z_k(u) |(j = 1,2, \ldots)\) 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.