  We consider whether L = limsup_{n to infty} n ||T^{n+1}-T^n|| < infty implies
that the operator T is power bounded. We show that this is so if L<1/e, but it
does not necessarily hold if L=1/e. As part of our methods, we improve a result
of Esterle, showing that if sigma(T) = {1} and T != I, then liminf_{n to infty}
n ||T^{n+1}-T^n|| >= 1/e. The constant 1/e is sharp. Finally we describe a way
to create many generalizations of Esterle's result, and also give many
conditions on an operator which imply that its norm is equal to its spectral
radius.
