Black Hole Mass Estimates and Rapid Growth of Supermassive Black Holes
  in Luminous $z \sim$ 3.5 Quasars
  We present new near-infrared (IR) observations of the H$\beta\ \lambda4861$
and MgII $\lambda2798$ lines for 32 luminous quasars with $3.2<z<3.9$ using the
Palomar Hale 200 inch telescope and the Large Binocular Telescope. We find that
the MgII Full Width at Half Maximum (FWHM) is well correlated with the H$\beta$
FWHM, confirming itself as a good substitute for the H$\beta$ FWHM in the black
hole mass estimates. The continuum luminosity at 5100 \AA\ well correlates with
the continuum luminosity at 3000 \AA\ and the broad emission line luminosities
(H$\beta$ and MgII). With simultaneous near-IR spectroscopy of the H$\beta$ and
MgII lines to exclude the influences of flux variability, we are able to
evaluate the reliability of estimating black hole masses based on the MgII line
for high redshift quasars. With the reliable H$\beta$ line based black hole
mass and Eddington ratio estimates, we find that the $z\sim3.5$ quasars in our
sample have black hole masses $1.90\times10^{9} M_{\odot} \lesssim M_{\rm BH}
\lesssim 1.37\times10^{10} M_{\odot}$, with a median of $\sim 5.14\times10^{9}
M_{\odot}$ and are accreting at Eddington ratios between 0.30 and 3.05, with a
median of $\sim1.12$. Assuming a duty cycle of 1 and a seed black hole mass of
$10^{4} M_{\odot}$, we show that the $z\sim3.5$ quasars in this sample can grow
to their estimated black hole masses within the age of the Universe at their
redshifts.
