Lower order and limiting directions of Julia sets of solutions to second order differential equations

Abstract

In this paper, we consider the properties of entire solutions to second order differential equation(⁎)f″+Af′+Bf=0, where A(z) and B(z)≢0 are entire functions. Under certain assumptions on A(z) and B(z), we prove that every non-trivial solution f of equation (*) is of infinite lower order, and then obtain the measure estimation of the limiting directions of Julia sets for those infinite lower order entire solutions. The existence of Baker domain for f(n) is also discussed.