  The effect of an hypothetical maximal proper acceleration on the inertial
mass of charged particles is investigated in the context of particle
accelerators. In particular, it is shown that maximal acceleration implies a
reduction of the luminosity of the bunches relative with respect to the
expected luminosity in the relativistic models of the bunches. This relative
loss in luminosity is of the order $10^{-7}$ for the LHC and of order $10^{-4}$
for current laser plasma accelerators facilities. Although this effect is
small, it increases with the square of the number of particles in the bunch.
