  Relations linking the temporal or/and spectral properties of the prompt
emission of gamma-ray bursts (hereafter GRBs) to the absolute luminosity are of
great importance as they both constrain the radiation mechanisms and represent
potential distance indicators. Here we discuss two such relations: the
lag-luminosity relation and the newly discovered duration-luminosity relation
of GRB pulses. We aim to extend our previous work on the origin of spectral
lags, using the duration-luminosity relation recently discovered by Hakkila et
al. to connect lags and luminosity. We also present a way to test this relation
which has originally been established with a limited sample of only 12 pulses.
We relate lags to the spectral evolution and shape of the pulses with a linear
expansion of the pulse properties around maximum. We then couple this first
result to the duration-luminosity relation to obtain the lag-luminosity and
lag-duration relations. We finally use a Monte-Carlo method to generate a
population of synthetic GRB pulses which is then used to check the validity of
the duration-luminosity relation. Our theoretical results for the lag and
duration-luminosity relations are in good agreement with the data. They are
rather insensitive to the assumptions regarding the burst spectral parameters.
Our Monte Carlo analysis of a population of synthetic pulses confirms that the
duration-luminosity relation must be satisfied to reproduce the observational
duration-peak flux diagram of BATSE GRB pulses. The newly discovered
duration-luminosity relation offers the possibility to link all three
quantities: lag, duration and luminosity of GRB pulses in a consistent way.
Some evidence for its validity have been presented but its origin is not easy
to explain in the context of the internal shock model.
