  We calculate the physical conditions and the N/H and O/H relative abundances
for a sample of long GRB (LGRB) host galaxies in the redshift range 1<z<2.1 by
modelling recently observed line and continuum spectra. The results are
consistent with those previously calculated for LGRB host galaxies throughout a
more extended redshift range z<3. We analyse star formation rates (SFR) within
the LGRB hosts on the basis of the Ha fluxes. They are compared with those of
local low-luminosity starburst (SB) galaxies, individual HII regions in local
galaxies as well as LGRB host galaxies at intermediate and relatively high
redshifts. The enhanced SFR in the HII regions within nearby galaxies is
explained by a relatively high filling factor which characterizes "individual
regions" rather than "entire galaxies" which are generally presented by the
observations. The fragmented matter in the galaxies derives from progenitor
merging. We check whether the release by the morphological transformations of
ice of the O_2 and N_2 molecules trapped into the ice mantles of dust grains
could explain the N/O ratios throughout the redshift. We have found that shock
velocities calculated by modelling the spectra are high enough to completely
destroy the ice mantles. Therefore, the prevention of secondary nitrogen
formation is a valid hypothesis to explain the low N/O ratios at z<1. The SFR
trend increasing with z is roughly similar to that of N/O.
