  We study in details how the pulse width of gamma-ray bursts is related with
energy under the assumption that the sources concerned are in the stage of
fireballs. Due to the Doppler effect of fireballs, there exists a power law
relationship between the two quantities within a limited range of frequency.
The power law range and the power law index depend strongly on the observed
peak energy $E_p$ as well as the rest frame radiation form, and the upper and
lower limits of the power law range can be determined by $E_p$. It is found
that, within the same power law range, the ratio of the $FWHM$ of the rising
portion to that of the decaying phase of the pulses is also related with energy
in the form of power laws. A platform-power-law-platform feature could be
observed in the two relationships. In the case of an obvious softening of the
rest frame spectrum, the two power law relationships also exist, but the
feature would evolve to a peaked one. Predictions on the relationships in the
energy range covering both the BATSE and Swift bands for a typical hard burst
and a typical soft one are made. A sample of FRED (fast rise and exponential
decay) pulse bursts shows that 27 out of the 28 sources belong to either the
platform-power-law-platform feature class or the peaked feature group,
suggesting that the effect concerned is indeed important for most of the
sources of the sample. Among these bursts, many might undergo an obvious
softening evolution of the rest frame spectrum.
