  We make a detailed pulse-wise study of gamma-ray bursts (GRBs) with known
redshift detected by \emph{Fermi}/Gamma Ray Burst Monitor (GBM). The sample
contains 19 GRBs with 43 pulses. We find that the average peak energy is
correlated to the radiated energy (the Amati relation) for individual pulses
with a correlation coefficient of 0.86, which is slightly better than the
correlation for the full GRBs. As the present correlation holds within GRBs, it
is a strong evidence supporting the reliability of such a correlation. We
investigate several aspects of this correlation. (i) We divide our sample into
redshift bins and study the evolution of the correlation. Though there is a
marginal indication of evolution of the correlation, we can conclude that the
present data is consistent with no evolution. (ii) We compare the correlation
in the first or single pulses of these GRBs to that of the rest of the pulses,
and confirm that the correlation is unaffected by the fact that first/single
pulses are generally harder than the rest. Finally, we conclude that the
pulse-wise Amati correlation is more robust and it has the potential of
refining the correlation so that GRB study could be used as a cosmological
tool.
