Measuring Supermassive Black Hole Masses: Correlation between the
  Redshifts of the Fe III UV Lines and the Widths of Broad Emission Lines
  We test the recently proposed (Mediavilla et al. 2018) black hole mass
scaling relationship based on the redshift {with respect to the quasar's rest
frame} of the Fe III$\lambda\lambda$2039-2113 line blend. To this end, we fit
this feature in the spectra of a well suited sample of quasars, observed with
X-shooter at the Very Large Telescope (VLT), whose masses have been
independently estimated using the virial theorem. For the quasars of this
sample we consistently confirm the redshift of the Fe
III$\lambda\lambda$2039-2113 blend and find that it correlates with the squared
widths of H$\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$\lambda\lambda$2039-2113 redshift based
masses are 0.18$\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$\lambda\lambda$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.
