  The Humphreys-Davidson (HD) limit sets the boundary between evolutionary
channels of massive stars that either end their lives as red supergiants (RSGs)
or as the hotter blue supergiants (BSGs) and Wolf-Rayet stars. Mixing in the
envelopes of massive stars close to their Eddington limit is crucial for
investigating the upper luminosity limit of the coolest supergiants. We study
the effects of excess mixing in superadiabatic layers that are dominated by
radiation pressure, and we critically investigate the effects of mixing and
mass loss on the evolution of RSGs with log (Teff/K) < 3.68 - as a function of
metallicity. Using MESA, we produce grids of massive star models at three
metallicities: Galactic (Zsol), LMC (1/2 Zsol) and SMC (1/5 Zsol), with both
high and low amounts of overshooting to study the upper luminosity limit of
RSGs. We systematically study the effects of excess mixing in the
superadiabatic layers of post-main sequence massive stars, overshooting above
the hydrogen core and yellow supergiant (YSG) mass-loss rates on the fraction
of core helium burning time spent as a RSG. We find that the excess mixing in
the superadiabatic layers is stronger at lower metallicities, as it depends on
the opacities in the hydrogen bump at log (Teff/K) ~ 4, which become more
pronounced at lower metallicity. This shifts the cutoff luminosities to lower
values at lower metallicities, thus balancing the first-order effect of mass
loss. The opposing effects of mass loss and excess envelope mixing during
post-main sequence evolution of stars with higher overshooting potentially
results in a metallicity-independent upper luminosity limit.
