We present a detailed model of an extreme slim disk ( ESD ) with a hot corona around a massive black hole with dimensionless accretion rate \dot { m } in the range 2.5 \ll \dot { m } \leq 100 .
We assume that a fraction f of the gravitational energy is released in the hot corona and the rest is released in the ESD .
The energy equation of the ESD is dominated by advection and the spectrum shows a broad “ hump ” caused by saturated Comptonization with monochromatic luminosity given by \nu L _ { \nu } \propto \nu ^ { 0 } .
This relationship enables us to estimate the black hole mass from the ESD luminosity .
The spectrum of the hot corona is sensitive to the parameter f \dot { M } and the viscosity and shows a Comptonized power-law with a high-energy cutoff .
The model is used to explain the spectral properties of narrow line Seyfert 1 galaxies ( NLS1s ) .
In particular , it can explain the spectrum of extreme NLS1s like NLS1 RE J1034+396 .
Our spectral estimate of the black hole mass in this source is in good agreement with the mass obtained by applying the reververation mapping correlation .
We also examine the Eddington ratios in a large NLS1 sample and find that most objects show super-Eddington accretion rates .
We argue that soft X-ray humps in NLS1s are natural consequences of super-Eddington accretion in such objects .