On the growth and the zeros of solutions of higher order linear differential equations with meromorphic coefficients

Abstract

We investigate the growth of meromorphic solutions of homoge-neous and nonhomogeneous higher order linear differential equationsf(k)+k−1∑j=1Ajf(j)+A0f= 0 (k>2),f(k)+k−1∑j=1Ajf(j)+A0f=Ak(k>2),whereAj(z) (j= 0,1, . . . , k) are meromorphic functions with finite order.Under some conditions on the coefficients, we show that all meromorphic so-lutionsf6≡0 of the above equations have an infinite order and infinite lowerorder. Furthermore, we give some estimates of their hyper-order, exponentand hyper-exponent of convergence of distinct zeros. We improve the resultsdue to Kwon; Chen and Yang; Belaïdi; Chen; Shen and Xu.