Some precise estimates of the hyper order of solutions of some complex linear differential equations

Abstract

Letρ(f)andρ2(f)denote respectively the order and the hyper order of an entirefunctionf.In this paper, we obtain some precise estimates of the hyper order ofsolutions of the following higher order linear differential equationsf(k)+k−1∑j=0Aj(z)ePj(z)f(j)= 0andf(k)+k−1∑j=0(Aj(z)ePj(z)+Bj(z))f(j)= 0wherek≥2,Pj(z) (j= 0,...,k−1)are nonconstantpolynomialssuchthatdegPj=n(j= 0,...,k−1)andAj(z) (6≡0),Bj(z) (6≡0)(j= 0,...,k−1)are entire functions withρ(Aj)<n,ρ(Bj)<n(j= 0,...,k−1). Under some conditions, we prove that every solutionf(z)6≡0of the above equations is of infinite order andρ2(f) =n.