We present near-infrared spectra of ten luminous , intermediate redshift quasars ( z \simeq 2 ; L _ { bol } \simeq 10 ^ { 47 } erg s ^ { -1 } ) , observed with SofI at the NTT of ESO/La Silla .
With these rest-frame optical spectra we probe the H \beta – [ O iii ] emission line region .
Using the standard scaling relation involving the width of the H \beta line and the continuum luminosity , we measure black hole masses in the range of \sim 2 \times 10 ^ { 9 } \raisebox { -2.15 pt } { $ \stackrel { < } { \sim } $ } M _ { bh } \raisebox { -2.15 pt } { $ \stackrel { < } { \sim } $ } 10 ^ { 10 } M _ { \odot } for these sources .
We also used SDSS spectra to probe Mg ii \lambda 2798 and C iv \lambda 1549 emission lines and used these for black hole mass measurements as well .
The black hole mass estimates using C iv \lambda 1549 are on average smaller by about 60 % than those based on H \beta .
The massive black holes we observe could not have grown by simple radiatively efficient accretion at the observed accretion rate starting from seeds of up to thousand solar masses .
About 10 % of the observed black hole mass must have been accumulated by earlier merger events and radiatively inefficient accretion .
Radiatively efficient accretion would further grow these BHs to masses of several 10 ^ { 9 } M _ { \odot } in 2–3 e-folding times i.e. , in several 10 ^ { 8 } yrs .
This scenario is consistent with recent models of BH growth .
The H \beta -based Eddington luminosity ratios are in the range of \sim 0.2 to 0.7 , with an average of < L _ { bol } /L _ { edd } > = 0.39 \pm 0.05 .
The L _ { bol } /L _ { edd } ratio distribution follows a log-normal distribution which is consistent with prior studies of quasars with comparable luminosity .
We also find that the gas metallicity of the broad-line region is super-solar with \sim 3 Z/Z _ { \odot } , based on N iii ] \lambda 1750 /O iii ] \lambda 1663 and N v \lambda 1240 /C iv \lambda 1549 emission line ratios .
We find no correlation of the gas metallicity with the optical Fe ii emission line strength in our small sample , contrary to a recent suggestion .