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

Maamar Andasmas, Benharrat Belaïdi

Abstract: We investigate the growth of meromorphic solutions of homogeneous and nonhomogeneous higher order linear differential equations $$ f^{(k)}+\sum_{j=1}^{k-1}A_jf^{(j)}+A_0f=0 (k\geqslant 2), $$ $$ f^{(k)}+\sum_{j=1}^{k-1}A_jf^{(j)}+A_0f=A_k (k\geqslant 2), $$ where $A_j(z)$ ($j=0,1,\dots,k$) are meromorphic functions with finite order. Under some conditions on the coefficients, we show that all meromorphic solutions $f\not\equiv 0$ of the above equations have an infinite order and infinite lower order. Furthermore, we give some estimates of their hyper-order, exponent and hyper-exponent of convergence of distinct zeros. We improve the results due to Kwon; Chen and Yang; Bela\"{i}di; Chen; Shen and Xu.

Keywords: linear differential equations; meromorphic functions; order of growth; hyper-order; exponent of convergence of zeros; hyper-exponent of convergence of zeros

Classification (MSC2000): 34M10; 30D35

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