  A variance inequality for spin-flip systems is obtained using comparatively
weaker knowledge of relaxation to equilibrium based on coupling estimates for
single site disturbances. We obtain variance inequalities interpolating between
the Poincar\'e inequality and the uniform variance inequality, and a general
weak Poincar\'e inequality. For monotone dynamics the variance inequality can
be obtained from decay of the autocorrelation of the spin at the origin, i.e.,
from that decay we conclude decay for general functions. This method is then
applied to the low temperature Ising model, where the time-decay of the
autocorrelation of the origin is extended to arbitrary quasi-local functions.
