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An improved transformation of the full width at half maxima (FWHM) of the [OIII] 5007 line in AGNs to the stellar velocity dispersion, sigma, of the hostgalaxy is given. This significantly reduces the systematic errors in using the[O III] FWHM as a proxy for sigma. AGN black hole masses, M, estimated usingthe Dibai single epoch spectrum method, are combined with the new estimates ofsigma to give a revised AGN M-sigma relationship extending up to high masses.This shows that the masses of the most massive black holes are systematicallyhigher than predicted by extrapolation of M \propto sigma^4 to high masses.This supports recent suggestions that stellar dynamical masses of the mostmassive black holes have been systematically underestimated. The steepening ofthe M-sigma relationship is consistent with the absence of very high sigmagalaxies in the local universe and with the curvature of the Faber-Jacksonrelationship. There appears to be significantly less intrinsic scatter in theM-sigma relationship for galaxies with black hole masses > 10^9 solar masses.It is speculated that this is connected with the core elliptical versusextra-light elliptical dichotomy. The low scatter in the high end of theM-sigma relationship implies that the transformation proposed here and theDibai method are good indicators of stellar velocity dispersion and massrespectively. There is no evidence for evolution of the M-sigma relationshipover time.

An Improved [O III] Line Width to Stellar Velocity Dispersion  Calibration: Curvature, Scatter, and Lack of Evolution in the Black-Hole Mass  Versus Stellar Velocity Dispersion Relationship