The FWHM of gamma-ray burst ( GRB ) pulses is known to be related with energy by a power-law .
We wonder if the power-law index \alpha is related with the corresponding local pulse width FWHM _ { 0 } .
Seven FRED ( fast rise and exponential decay ) pulse GRBs are employed to study this issue , where six of them were interpreted recently by the relativistic curvature effect ( the Doppler effect of fireballs ) and the corresponding local pulses were intensely studied .
A regression analysis shows an anti-correlation between log \alpha and logFWHM _ { 0 } with a slope of -0.37 \pm 0.13 .
This suggests that , for the class of the GRB pulses which are consequences of the curvature effect , the difference of the local pulse width might lead to the variation of the power law index , where the smaller the width the larger the value of \alpha .
Since the number of sources employed in this analysis is small , our result is only a preliminary one which needs to be confirmed by larger samples .