Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 114, pp. 1-17.
Title: Growth of solutions to higher-order linear differential equations
with entire coefficients
Authors: Habib Habib (Univ.of Mostaganem, Algeria)
Benharrat Belaidi (Univ.of Mostaganem, Algeria)
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.