Hokkaido Mathematical Journal

Growth of meromorphic solutions of some linear differential equations

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

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

Digital Object Identifier
doi:10.14492/hokmj/1510045308

Mathematical Reviews number (MathSciNet)
MR3720339

Zentralblatt MATH identifier
1384.34091

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