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 ƒ(k) + Ak-1 (z)ƒ(k-1) + ··· + A1(z)ƒ' + A0(z)ƒ = 0, where A0(z),..., Ak-1(z) are entire functions with A0(z) ≢ 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 ≢ 0 will have infinite order and we estimate in one case that lower bounds of the hyper-order.