  Suppose that $f(z)$ is a transcendental entire function and that the Fatou
set $F(f)\neq\emptyset$. Set $$B_1(f):=\sup_{U}\frac{\sup_{z\in
U}\log(|z|+3)}{\inf_{w\in U}\log(|w|+3)}$$ and
$$B_2(f):=\sup_{U}\frac{\sup_{z\in U}\log\log(|z|+30)}{\inf_{w\in
U}\log(|w|+3)},$$ 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.
