  Let $f$ be a transcendental entire function. The quite fast escaping set,
$Q(f)$, and the set $Q_2(f),$ which was defined recently, are equal to the fast
escaping set, $A(f),$ under certain conditions. In this paper we generalise
these sets by introducing a family of sets $Q_m(f)$, $m \in \mathbb{N}.$ We
also give one regularity and one growth condition which imply that $Q_m(f)$ is
equal to $A(f)$ and we show that all functions of finite order and positive
lower order satisfy $Q_m(f)=A(f)$ for any $m$. Finally, we relate the new
regularity condition to a sufficient condition for $Q_2(f)=A(f)$ introduced in
recent work.
