Uniformly bounded components of normality

Abstract

Suppose that $f(z)$ is a transcendental entire function and that the Fatou set $F(f)\ne \varnothing$. Set $$B_1(f):=\underset{U}{\sup }\frac{{\sup }_{z\in U}\log (|z|+3)}{{\inf }_{w\in U}\log (|w|+3)}$$ and $$B_2(f):=\underset{U}{\sup }\frac{{\sup }_{z\in U}\log \log (|z|+30)}{{\inf }_{w\in U}\log (|w|+3)},$$ where the supremum ${\sup }_U$ is taken over all components of $F(f)$. If $B_1(f)<\infty$ or $B_2(f)<\infty$, then we say $F(f)$ is strongly uniformly bounded or uniformly bounded respectively. In this article, we will show that, under some conditions, $F(f)$ is (strongly) uniformly bounded.

17 pages, a revised version, to appear in Mathematical Proceedings Cambridge Philosophical Society