Composite entire functions with no unbounded Fatou components

Let L be the set of all entire functions f such that for given >0, log L(r, f ) > (1− ) logM(r,f ) for all r outside a set of logarithmic density zero. Let F = K 1 FK where FK is the set of all transcendental entire functions f such that log logM(r,f ) (log r) 1 K . If h = fN ◦ fN−1 ◦ ··· ◦ f1 where fi ∈ F ∩ L (i = 1, . . . , N), then it is shown that h has no unbounded Fatou component. © 2007 Elsevier Inc. All rights reserved