Growth of solutions of an n-th order linear differential equation with entire coefficients

Abstract

We consider a differential equation f (n)+ A n− 1 (z) f (n− 1)+…+ A 1 (z) f'+ A 0 (z) f= 0, where A 0 (z),..., A n− 1 (z) are entire functions with A 0 (z){¬≡} 0. Suppose that there exist a positive number μ, and a sequence (z j) j∈ N with lim j→+∞ z j=∞, and also two real numbers α, β (0≤ β< α) such that| A 0 (z j)|≥ e α| z j| μ and| A k (z j)|≤ e β| z j| μ as j→+∞(k= 1,..., n− 1). We prove that all solutions f {¬≡} 0 of this equation are of infinite order. This result is a generalization of one theorem of Gundersen ([3], p. 418).