  We obtain bounds on the decay exponent lambda of the autocorrelation function
in phase ordering dynamics. For non-conserved order parameter, we recover the
Fisher and Huse inequality, lambda > = d/2. If the order parameter is conserved
we also find lambda >= d/2 if the initial time t1 = 0. However, for t1 in the
scaling regime, we obtain lambda >= d/2 + 2 for d >= 2 and lambda >= 3/2 for
d=1. For the one-dimensional scalar case, this, in conjunction with previous
results, implies that lambda is different for t1 = 0 and t1 >> 1. In
2-dimensions, our extensive numerical simulations for a conserved scalar order
parameter show that lambda approx 3 for t1=0 and lambda approx 4 for t1 >> 1.
These results contradict a recent conjecture that conservation of order
parameter requires lambda = d. Quenches to and from the critical point are also
discussed. (uuencoded PostScript file with figures appended at end).
