  Laboratory experiments measuring neutrino oscillations, indicate small mass
differences between different mass eigenstates of neutrinos. The absolute mass
scale is however not determined, with at present the strongest upper limits
coming from astronomical observations rather than terrestrial experiments. The
presence of massive neutrinos suppresses the growth of perturbations below a
characteristic mass scale, thereby leading to a decreased abundance of
collapsed dark matter halos. Here we show that this effect can significantly
alter the predicted luminosity function (LF) of high redshift galaxies. In
particular we demonstrate that a stringent constraint on the neutrino mass can
be obtained using the well measured galaxy LF and our semi-analytic structure
formation models. Combining the constraints from the Wilkinson Microwave
Anisotropy Probe 7 year (WMAP7) data with the LF data at z = 4, we get a limit
on the sum of the masses of 3 degenerate neutrinos \Sigma m_\nu < 0.52 eV at
the 95 % CL. The additional constraints using the prior on Hubble constant
strengthens this limit to \Sigma m_\nu < 0.29 eV at the 95 % CL. This neutrino
mass limit is a factor of order 4 improvement compared to the constraint based
on the WMAP7 data alone, and as stringent as known limits based on other
astronomical observations. As different astronomical measurements may suffer
from different set of biases, the method presented here provides a
complementary probe of \Sigma m_\nu . We suggest that repeating this exercise
on well measured luminosity functions over different redshift ranges can
provide independent and tighter constraints on \Sigma m_\nu .
