  We investigate whether the beta function of the finite-$N$ Gross-Neveu model,
as calculated up to the four-loop level, exhibits evidence for an infrared
zero. As part of our analysis, we calculate and analyze Pad\'e approximants to
this beta function and evaluate effects of scheme dependence. From our study,
we find that in the range of coupling where the perturbative calculation of the
four-loop beta function is reliable, it does not exhibit robust evidence for an
infrared zero.
