I have selected a sample of 504 HII like-starburst galaxies according to the equivalent widths in their emission lines in a range of redshift of 0.1 < z < 3.4 from the literature, in order to assess the validity of the L(HB) - g relation at high redshift and its use as an accurate distance estimator. The physical parameters for the sample at high redshift, within the uncertainties, are consistent with the ones for the HIIGx at low redshift. This suggests that the physical properties for HIIGx at low and high redshift are similar. By combining our sample at high redshift with the local sample of HIIGx from Chávez et al. (2014), I find an evident flattening that starts at approximately log(g) = 1:8 km s-1 in the L(HB) - g relation. This flattening has an interesting application since it leads to a kind of standard candle for those HIIGx with log(g) > 1:8 km s-1. This flattening or the existence of an upper limit in the luminosity of HIIGx has not been reported before. Applying this new HIIGx standard candle method to the high redshift sample we obtain good restrictions on the cosmological parameters solution space. The combination of our sample at high redshift with the local sample of 156 (HIIGx and GEHRs) dramatically improves the constraints on the plane (nm, w0), which are consistent with the results from SNeIa. These results are surprisingly good considering the high uncertainties in the data at high redshift, therefore we expect better constraints on the plane (nm, w0) using high quality data from high resolution spectrographs at 8 m class telescopes.