Limiting direction of Julia sets and infinite radial order of solutions to complex linear differential equations

Abstract

In this paper we find that for infinite order entire functions, the ray where it takes infinite radial order is a common limiting directions of Julia sets of their derivatives and their primitives.  Applying this result to the solutions of some complex differential equations, we obtain the lower bound of the measure of sets of common limiting directions of Julia sets of the derivatives and integral primitives of any non-trivial solution of these equations, which give alternative proofs of previous results.