Composite entire functions with no unbounded Fatou components

Abstract

Let L be the set of all entire functions f such that for given ϵ > 0 , log L ( r , f ) > ( 1 − ϵ ) log M ( r , f ) for all r outside a set of logarithmic density zero. Let F = ⋃ K ⩾ 1 F K where F K is the set of all transcendental entire functions f such that log log M ( r , f ) ⩾ ( log r ) 1 K . If h = f N ○ f N − 1 ○ ⋯ ○ f 1 where f i ∈ F ∩ L ( i = 1 , … , N ) , then it is shown that h has no unbounded Fatou component.