Electronic Journal of Differential Equations (Feb 2003)
Order and hyper-order of entire solutions of linear differential equations with entire coefficients
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 $A_{0}(z) otequiv 0$. 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 $f otequiv 0$ will have infinite order and we estimate in one case the lower bounds of the hyper-order.
