Growth Conditions for Entire Functions with Only Bounded Fatou Components

Let f be a transcendental entire function of order < 1/2. We denote the maximum and minimum modulus of f by M(r, f) = max{|f(z)| : |z| = r} and m(r, f) = min{|f(z)| : |z| = r}. We obtain a minimum modulus condition satisfied by many f of order zero that implies all Fatou components are bounded. A special case of our result is that if log logM(r, f) = O(log r/(log log r)) for some K > 1, then there exist α > 1 and C > 0 such that for all large R, there exists r ∈ (R,R] with logm(r, f) logM(R, f) ≥ α ( 1− C (log logR)K ) , and this in turn implies boundedness of all Fatou components. The condition on m(r, f) is a refined form of a minimum modulus conjecture formulated by the first author. We also show that there are some functions of order zero, and there are functions of any positive order, for which even refined forms of the minimum modulus conjecture fail. Our results and counterexamples indicate rather precisely the limits of the method of using the minimum modulus to rule out the existence of unbounded Fatou components.