Uniformly bounded components of normality

Abstract

Suppose that f(z) is a transcendental entire function and that the Fatou set F(f) ‡ ∅. Set B1(f):=supU sup z∈U log(|z| + 3)/infw∈U log(|w| + 3) and B 2(f):=supU supz∈U log(|z| + 30)/inf w∈U log(|w| + 3), where the supremum supU is taken over all components of F(f). If B1(f) < ∞ or B 2(f) < ∞, then we say F(f) is strongly uniformly bounded or uniformly bounded respectively. We show that, under some conditions, F(f) is (strongly) uniformly bounded. © 2007 Cambridge Philosophical Society.