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

Let P(z) and $\tilde{P}\left(z\right)$ be polynomials of the same degree. We consider the equations u″ = P(z)u and ${\tilde{u}}^{\prime \prime }=\tilde{P}\left(z\right)\tilde{u}\text{(}z\in ℂ\text{)}$ whose solutions are u(z) and $\tilde{u}\left(z\right)$, respectively. Let z_k(u) and $z_k\left(\tilde{u}\right),k=1,2,\mathrm{\dots },$ be the zeros of u(z) and $\tilde{u}\left(z\right)$, respectively. We derive bounds for the quantity

${}_k^{inf}\left|\frac{1}{z_k\left(u\right)}-\frac{1}{z_j\left(\tilde{u}\right)}\right|.$