Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 17, pp. 1-12.
Title: Order and hyper-order of entire solutions of linear
differential equations with entire coefficients
Authors: Benharrat Belaidi (University of Mostaganem, Algeria)
Karima Hamani (University of Mostaganem, Algeria)
Abstract:
In this paper, we investigate the growth of solutions of
the differential equation
$$
f^{(k)}+A_{k-1}( z)f^{( k-1) }+\dots+A_{1}( z) f'+A_{0}(z) f=0,
$$
where $A_{0}( z) ,\dots, A_{k-1}(z)$
are entire functions with $A_{0}(z) \not\equiv 0$.
We will show that if the coefficients satisfy certain
growth conditions, then every finite order solution of the equation will
satisfy certain other growth conditions. We will also find conditions on the
coefficients so that every solution $f\not\equiv 0$ will have infinite order
and we estimate in one case the lower bounds of the hyper-order.
Submitted November 27, 2002. Published February 20, 2003.
Math Subject Classifications: 30D35, 34M10, 34C11.
Key Words: Linear differential equations; growth of entire functions;
hyper-order.