Bounds for the sums of zeros of solutions of $u^{(m)}=P(z)u$ where $P$ is a polynomial

Abstract

The main purpose of this paper is to consider the differential equation $u^{(m)}=P(z)u$ $(m\geq 2)$ where $P$ is a polynomial with in general complex coefficients. Let $z_{k}(u),$ $k=1,2,\ldots$ be the zeros of a nonzero solution $u$ to that equation. We obtain bounds for the sums $$\sum_{k=1}^{j}\frac{1}{|z_{k}(u)|}\quad (j\in\mathbb{N})$$ which extend some recent results proved by Gil'.

Keywords

• complex differential equation
• zeros of solution
• polynomial