  The stellar initial mass function (IMF) is expressed by $\phi(m) \propto
m^{-\alpha}$ with the slope $\alpha$, and known as the poorly-constrained but
very important function in studies of star and galaxy formation. There are no
sensible observational constraints on the IMF slopes beyond Milky Way and
nearby galaxies. Here we combine two sets of observational results, 1) cosmic
densities of core-collapse supernova explosion (CCSNe) rates and 2) cosmic far
ultraviolet radiation (and infrared re-radiation) densities, which are
sensitive to massive ($\simeq 8-50 \,{\rm M}_\odot$) and moderately massive
($\simeq 2.5-7 \,{\rm M}_\odot$) stars, respectively, and constrain the IMF
slope at $m>1\,{\rm M}_\odot$ with a freedom of redshift evolution. Although no
redshift evolution is identified beyond the uncertainties, we find that the
cosmic average IMF slope at $z=0$ is $\alpha=1.8-3.2$ at the 95 % confidence
level that is comparable with the Salpeter IMF, $\alpha=2.35$, which marks the
first constraint on the cosmic average IMF. We show a forecast for the Nancy
Grace Roman Space Telescope supernova survey that will provide significantly
strong constraints on the IMF slope with $\delta \alpha\simeq 0.5$ over
$z=0-2$. Moreover, as for an independent IMF probe instead of 1), we suggest to
use diffuse supernovae neutrino background (DSNB), relic neutrinos from CCSNe.
We expect that the Hyper-Kamiokande neutrino observations over 20 years will
improve the constraints on the IMF slope and the redshift evolution
significantly better than those obtained today, if the systematic uncertainties
of DSNB production physics are reduced in the future numerical simulations.
