## International Journal of Differential Equations

- Int. J. Differ. Equ.
- Volume 2015 (2015), Article ID 690519, 6 pages.

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

#### Abstract

We consider the equation ${u}^{\u2033}=P(z)u+F(z)\hspace{0.17em}\hspace{0.17em}(z\in \mathbb{C})$, 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.

#### Article information

**Source**

Int. J. Differ. Equ., Volume 2015 (2015), Article ID 690519, 6 pages.

**Dates**

Received: 14 July 2015

Revised: 22 October 2015

Accepted: 29 October 2015

First available in Project Euclid: 20 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.ijde/1484881449

**Digital Object Identifier**

doi:10.1155/2015/690519

**Mathematical Reviews number (MathSciNet)**

MR3429206

**Zentralblatt MATH identifier**

1339.34094

#### Citation

Gil’, Michael. Bounds for Products of Zeros of Solutions to Nonhomogeneous ODE with Polynomial Coefficients. Int. J. Differ. Equ. 2015 (2015), Article ID 690519, 6 pages. doi:10.1155/2015/690519. https://projecteuclid.org/euclid.ijde/1484881449