We test the recently proposed (Mediavilla et al. 2018) black hole mass scaling relationship based on the redshift {with respect to the quasars rest frame} of the Fe III$lambdalambda$2039-2113 line blend. To this end, we fit this feature in the spectra of a well suited sample of quasars, observed with X-shooter at the Very Large Telescope (VLT), whose masses have been independently estimated using the virial theorem. For the quasars of this sample we consistently confirm the redshift of the Fe III$lambdalambda$2039-2113 blend and find that it correlates with the squared widths of H$beta$, H$alpha$ and Mg II, which are commonly used as a measure of $M_{BH}/R$ to determine masses from the virial theorem. The average differences between virial and Fe III$lambdalambda$2039-2113 redshift based masses are 0.18$pm 0.21$ dex, 0.18$pm 0.22$ dex and 0.14$pm 0.21$ dex, when the full widths at half maximum (FWHM) of the H$beta$, H$alpha$ and MgII lines are, respectively, used. The difference is reduced to 0.10$pm 0.16$ dex when the standard deviation, $sigma$, of {the} MgII line is used, instead. We also study the high S/N composite quasar spectra of the Baryon Oscillation Spectroscopic Survey (BOSS), finding that the Fe III$lambdalambda$2039-2113 redshifts and Mg II squared widths, $FWHM_{MgII}^2$, match very well the correlation found for the individual quasar spectra observed with X-shooter. This correlation is expected if the redshift is gravitational.
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