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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.