## Hokkaido Mathematical Journal

- Hokkaido Math. J.
- Volume 39, Number 1 (2010), 127-138.

### Growth of solutions and oscillation of differential polynomials generated by some complex linear differential equations

Benharrat BELAÏDI and Abdallah EL FARISSI

#### Abstract

This paper is devoted to studying the growth and the oscillation of solutions of the second order non-homogeneous linear differential equation $$ f''+A_{1} (z) e^{P (z)}f'+A_{0} (z) e^{Q (z)}f = F, $$ where $P (z)$, $Q (z)$ are nonconstant polynomials such that $\deg P=\deg Q=n$ and $A_{j} (z)$ $( \not\equiv 0 )$ $(j=0,1)$, $F\not\equiv 0$ are entire functions with $\rho ( A_{j} ) < n$ $( j=0,1 )$. We also investigate the relationship between small functions and differential polynomials $g_{f} (z)=d_{2}f''+d_{1}f'+d_{0}f$, where $d_{0} (z)$, $d_{1} (z)$, $d_{2} (z)$ are entire functions that are not all equal to zero with $\rho ( d_{j} ) < n$ $( j=0,1,2 )$ generated by solutions of the above equation.

#### Article information

**Source**

Hokkaido Math. J., Volume 39, Number 1 (2010), 127-138.

**Dates**

First available in Project Euclid: 19 May 2010

**Permanent link to this document**

https://projecteuclid.org/euclid.hokmj/1274275023

**Digital Object Identifier**

doi:10.14492/hokmj/1274275023

**Mathematical Reviews number (MathSciNet)**

MR2649330

**Zentralblatt MATH identifier**

1201.34136

**Subjects**

Primary: 34M10: Oscillation, growth of solutions

Secondary: 30D35: Distribution of values, Nevanlinna theory

**Keywords**

linear differential equations entire solutions order of growth exponent of convergence of zeros exponent of convergence of distinct zeros

#### Citation

BELAÏDI, Benharrat; EL FARISSI, Abdallah. Growth of solutions and oscillation of differential polynomials generated by some complex linear differential equations. Hokkaido Math. J. 39 (2010), no. 1, 127--138. doi:10.14492/hokmj/1274275023. https://projecteuclid.org/euclid.hokmj/1274275023