Composite entire functions with no unbounded Fatou components

Let LL be the set of all entire functions f   such that for given ϵ>0ϵ>0,logL(r,f)>(1−ϵ)logM(r,f)logL(r,f)>(1−ϵ)logM(r,f) for all r   outside a set of logarithmic density zero. Let F=⋃K⩾1FKF=⋃K⩾1FK where FKFK is the set of all transcendental entire functions f   such that loglogM(r,f)⩾(logr)1K. If h=fN○fN−1○⋯○f1h=fN○fN−1○⋯○f1 where fi∈F∩Lfi∈F∩L(i=1,…,N)(i=1,…,N), then it is shown that h has no unbounded Fatou component.