Electron. J. Diff. Equ., Vol. 2014 (2014), No. 114, pp. 1-17.

Growth of solutions to higher-order linear differential equations with entire coefficients

Habib Habib, Benharrat Belaidi

Abstract:
In this article, we discuss the order and hyper-order of the linear differential equation
$$ f^{(k) }+\sum_{j=1}^{k-1} (B_je^{b_jz}+D_je^{d_jz}) f^{(j) }+( A_1e^{a_1z}+A_2e^{a_2z}) f=0, $$
where $A_j(z), B_j(z), D_j(z)$ are entire functions $(\not\equiv 0)$ and $a_1,a_2,d_j$ are complex numbers $(\neq 0)$, and $b_j$ are real numbers. Under certain conditions, we prove that every solution $f\not\equiv 0$ of the above equation is of infinite order. Then, we obtain an estimate of the hyper-order. Finally, we give an estimate of the exponent of convergence for distinct zeros of the functions $f^{(j)}-\varphi $ $(j=0,1,2) $, where $\varphi$ is an entire function $(\not\equiv 0) $ and of order $\sigma (\varphi)<1$, while the solution f of the differential equation is of infinite order. Our results extend the previous results due to Chen, Peng and Chen and others.

Submitted November 22, 2013. Published April 21, 2014.
Math Subject Classifications: 34M10, 30D35.
Key Words: Linear differential equation; entire solution; order of growth; hyper-order of growth; fixed point.

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Habib Habib
Department of Mathematics
Laboratory of Pure and Applied Mathematics
University of Mostaganem (UMAB)
B. P. 227 Mostaganem, Algeria
email: habibhabib2927@yahoo.fr
Benharrat Belaïdi
Department of Mathematics
Laboratory of Pure and Applied Mathematics
University of Mostaganem (UMAB)
B. P. 227 Mostaganem, Algeria
email: belaidi@univ-mosta.dz

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