  We develop further the approach to upper and lower bounds in quantum dynamics
via complex analysis methods which was introduced by us in a sequence of
earlier papers. Here we derive upper bounds for non-time averaged outside
probabilities and moments of the position operator from lower bounds for
transfer matrices at complex energies. Moreover, for the time-averaged
transport exponents, we present improved lower bounds in the special case of
the Fibonacci Hamiltonian. These bounds lead to an optimal description of the
time-averaged spreading rate of the fast part of the wavepacket in the large
coupling limit. This provides the first example which demonstrates that the
time-averaged spreading rates may exceed the upper box-counting dimension of
the spectrum.
