  The galaxy stellar mass function (GSMF) at high-z provides key information on
star-formation history and mass assembly in the young Universe. We aimed to use
the unique combination of deep optical/NIR/MIR imaging provided by HST, Spitzer
and the VLT in the CANDELS-UDS, GOODS-South, and HUDF fields to determine the
GSMF over the redshift range 3.5<z<7.5. We utilised the HST WFC3/IR NIR imaging
from CANDELS and HUDF09, reaching H~27-28.5 over a total area of 369 arcmin2,
in combination with associated deep HST ACS optical data, deep Spitzer IRAC
imaging from the SEDS programme, and deep Y and K-band VLT Hawk-I images from
the HUGS programme, to select a galaxy sample with high-quality photometric
redshifts. These have been calibrated with more than 150 spectroscopic
redshifts in the range 3.5<z<7.5, resulting in an overall precision of
sigma_z/(1+z)~0.037. We have determined the low-mass end of the high-z GSMF
with unprecedented precision, reaching down to masses as low as M*~10^9 Msun at
z=4 and ~6x10^9 Msun at z=7. We find that the GSMF at 3.5<z<7.5 depends only
slightly on the recipes adopted to measure the stellar masses, namely the
photo-z, the SFHs, the nebular contribution or the presence of AGN on the
parent sample. The low-mass end of the GSMF is steeper than has been found at
lower redshifts, but appears to be unchanged over the redshift range probed
here. Our results are very different from previous GSMF estimates based on
converting UV galaxy luminosity functions into mass functions via tight M/L
relations. Integrating our evolving GSMF over mass, we find that the growth of
stellar mass density is barely consistent with the time-integral of the SFR
density over cosmic time at z>4. These results confirm the unique synergy of
the CANDELS+HUDF, HUGS, and SEDS surveys for the discovery and study of
moderate/low-mass galaxies at high redshifts.
