  We prove a form of the $\cos \pi \rho$ theorem which gives strong estimates
for the minimum modulus of a transcendental entire function of order zero. We
also prove a generalisation of a result of Hinkkanen that gives a sufficient
condition for a transcendental entire function to have no unbounded Fatou
components. These two results enable us to show that there is a large class of
entire functions of order zero which have no unbounded Fatou components. On the
other hand we give examples which show that there are in fact functions of
order zero which not only fail to satisfy Hinkkanen's condition but also fail
to satisfy our more general condition. We also give a new regularity condition
that is sufficient to ensure that a transcendental entire function of order
less than 1/2 has no unbounded Fatou components. Finally, we observe that all
the conditions given here which guarantee that a transcendental entire function
has no unbounded Fatou components, also guarantee that the escaping set is
connected, thus answering a question of Eremenko for such functions.

Complex Variables;Dynamical Systems