  Let $f$ be a transcendental entire function. The fast escaping set, $A(f)$,
plays a key role in transcendental dynamics. The quite fast escaping set,
$Q(f)$, defined by an apparently weaker condition is equal to $A(f)$ under
certain conditions. Here we introduce $Q_2(f)$ defined by what appears to be an
even weaker condition. Using a new regularity condition we show that functions
of finite order and positive lower order satisfy $Q_2(f)=A(f)$. We also show
that the finite composition of such functions satisfies $Q_2(f)=A(f)$. Finally,
we construct a function for which $Q_2(f) \neq Q(f)= A(f)$.
