We present new near-infrared ( IR ) observations of the H \beta \lambda 4861 and MgII \lambda 2798 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 Å well correlates with the continuum luminosity at 3000 Å 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 \sim 3.5 quasars in our sample have black hole masses 1.90 \times 10 ^ { 9 } M _ { \odot } \lesssim M _ { BH } \lesssim 1.37 \times 10 ^ { 10 } M _ { \odot } , with a median of \sim 5.14 \times 10 ^ { 9 } M _ { \odot } and are accreting at Eddington ratios between 0.30 and 3.05 , with a median of \sim 1.12 .
Assuming a duty cycle of 1 and a seed black hole mass of 10 ^ { 4 } M _ { \odot } , we show that the z \sim 3.5 quasars in this sample can grow to their estimated black hole masses within the age of the Universe at their redshifts .