Growth of solutions to higher-order linear differential equations with entire coefficients

Abstract

In this article, we discuss the order and hyper-order of the lineardifferential equationf(k)+k−1Xj=1(Bjebjz+Djedjz)f(j)+ (A1ea1z+A2ea2z)f= 0,whereAj(z),Bj(z),Dj(z) are entire functions (6≡0) anda1,a2,djare complexnumbers (6= 0), andbjare real numbers. Under certain conditions, we provethat every solutionf6≡0 of the above equation is of infinite order. Then,we obtain an estimate of the hyper-order. Finally, we give an estimate of theexponent of convergence for distinct zeros of the functionsf(j)−φ(j= 0,1,2),whereφis an entire function (6≡0) and of orderσ(φ)<1, while the solutionfof the differential equation is of infinite order. Our results extend the previousresults due to Chen, Peng and Chen and others.