## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 46, Number 3 (2017), 487-512.

### Growth of meromorphic solutions of some linear differential equations

Hamid BEDDANI and Karima HAMANI

#### Abstract

In this paper, we investigate the order and the hyper-order of meromorphic solutions of the linear differential equation \begin{equation*} f^{(k)}+\sum^{k-1}_{j=1}(D_{j}+B_{j}e^{P_{j}(z) })f^{(j)}+( D_{0}+A_{1}e^{Q_{1}( z)}+A_{2}e^{Q_{2}( z) }) f=0, \end{equation*} where $k\geq 2$ is an integer, $Q_{1}(z),Q_{2}(z)$, $P_{j}(z) $ $(j=1, \dots ,k-1)$ are nonconstant polynomials and $A_{s}(z)$ $(\not\equiv 0)$ $(s=1,2)$, $B_{j}( z)$ $(\not\equiv 0)$ $(j=1, \dots ,k-1)$, $D_{m}(z)$ $(m=0,1, \dots ,k-1)$ are meromorphic functions. Under some conditions, we prove that every meromorphic solution $f$ $(\not\equiv 0)$ of the above equation is of infinite order and we give an estimate of its hyper-order. Furthermore, we obtain a result about the exponent of convergence and the hyper-exponent of convergence of a sequence of zeros and distinct zeros of $f-\varphi$, where $\varphi$ $(\not\equiv 0)$ is a meromorphic function and $f$ $(\not\equiv 0)$ is a meromorphic solution of the above equation.

#### Article information

**Source**

Hokkaido Math. J., Volume 46, Number 3 (2017), 487-512.

**Dates**

First available in Project Euclid: 7 November 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1510045308

**Digital Object Identifier**

doi:10.14492/hokmj/1510045308

**Mathematical Reviews number (MathSciNet)**

MR3720339

**Zentralblatt MATH identifier**

1384.34091

**Subjects**

Primary: 34M10: Oscillation, growth of solutions 30D35: Distribution of values, Nevanlinna theory

**Keywords**

Linear Differential Equation Meromorphic function Hyper-order Exponent of convergence hyper-exponent of convergence

#### Citation

BEDDANI, Hamid; HAMANI, Karima. Growth of meromorphic solutions of some linear differential equations. Hokkaido Math. J. 46 (2017), no. 3, 487--512. doi:10.14492/hokmj/1510045308. https://projecteuclid.org/euclid.hokmj/1510045308