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

Abstract EN

We consider the equation u ″ = P ( z ) u + F ( z ) ( z ∈ C ) , where P ( z ) is a polynomial and F ( z ) is an entire function.

Let z k ( u ) ( k = 1,2 , … ) be the zeros of a solution u ( z ) to that equation.

Lower estimates for the products ∏ k = 1 j | z k ( u ) | ( j = 1,2 , … ) 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.