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October 2017 Growth of meromorphic solutions of some linear differential equations
Hamid BEDDANI, Karima HAMANI
Hokkaido Math. J. 46(3): 487-512 (October 2017). DOI: 10.14492/hokmj/1510045308

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.

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Hamid BEDDANI. Karima HAMANI. "Growth of meromorphic solutions of some linear differential equations." Hokkaido Math. J. 46 (3) 487 - 512, October 2017. https://doi.org/10.14492/hokmj/1510045308

Information

Published: October 2017
First available in Project Euclid: 7 November 2017

zbMATH: 1384.34091
MathSciNet: MR3720339
Digital Object Identifier: 10.14492/hokmj/1510045308

Subjects:
Primary: 30D35, 34M10

Rights: Copyright © 2017 Hokkaido University, Department of Mathematics

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Vol.46 • No. 3 • October 2017
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