On the growth of solutions of some higher order linear differential equations with entire coefficients

Abstract

In this paper, we investigate the order and the hyper-order of solutions of the linear differential equation f (k) +

Dk−1 + Bk−1e bk−1z

f (k−1) + ... +

D1 + B1e b1z

f ′ + (D0 + A1e a1z + A2e a2z ) f = 0, where Aj (z) (6≡ 0) (j = 1, 2), Bl (z) (6≡ 0) (l = 1, ..., k − 1), Dm (m = 0, ..., k − 1) are entire functions with max{σ (Aj ), σ (Bl), σ (Dm)} < 1, a1, a2, bl (l = 1, ..., k − 1) are complex numbers. Under some conditions, we prove that every solution f (z) 6≡ 0 of the above equation is of infinite order and with hyper-order