  Let $f$ be a transcendental entire function. The fast escaping set $A(f)$,
various regularity conditions on the growth of the maximum modulus of $f$, and
also, more recently, the quite fast escaping set $Q(f)$ have all been used to
make progress on fundamental questions concerning the iteration of $f$. In this
paper we establish new relationships between these three concepts.
  We prove that a certain weak regularity condition is necessary and sufficient
for $Q(f)=A(f)$ and give examples of functions for which $Q(f)\neq A(f)$.
  We also apply a result of Beurling that relates the size of the minimum
modulus of $f$ to the growth of its maximum modulus in order to establish that
a stronger regularity condition called log-regularity holds for a large class
of functions, in particular for functions in the Eremenko-Lyubich class
${\mathcal B}$.
