## International Journal of Differential Equations

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

Michael Gil’

#### Abstract

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 ${z}_{k}(u) (k=\mathrm{1,2},\dots )$ be the zeros of a solution $u(z)$ to that equation. Lower estimates for the products ${\prod }_{k=\mathrm{1}}^{j}|{z}_{k}(u)| (j=\mathrm{1,2},\dots )$ 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
Revised: 22 October 2015
Accepted: 29 October 2015
First available in Project Euclid: 20 January 2017

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

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