Fast escaping points of entire functions: a new regularity condition

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 $Q_2(f) \neq Q(f)= A(f)$.