Infinite order solutions of complex linear differential equations

Abstract

In this paper we investigate the growth of solutions of the differential equation f .k/C Akð€€€1 .´/f .kð€€€1/C CA1 .´/f 0 CA0 .´/f D 0; where A0 .´/ ; : : : ; Akð€€€1 .´/ are entire functions with 0 < .A0/ 1=2: We will show that if there exists a real constant < .A0/ and a set E .1;C1/ with log densE D 1; such that for all r 2 E ; we have minj´jDr jAj .´/ j exp.r / .j D 1;2; : : : ;k ð€€€1/ ; then every solution f 6 0 of the above differential equation is of infinite order with hyper-order 2 .f / .A0/. The paper extends previous results by the author and Hamani.