  (abbreviated) We show that there is a limit relation between the black hole
mass and the width at the half maximum of H$\beta$ for active galactic nuclei
(AGNs) with super-Eddington accretion rates. When a black hole has a
super-Eddington accretion rate, the empirical relation of reverberation mapping
has two possible ways. First, it reduces to a relation between the black hole
mass and the size of the broad line region due to the photon trapping effects
inside the accretion disk. For the Kaspi et al.'s empirical reverberation
relation, we get the limit relation as $M_{\rm BH}=(2.9 - 12.6)\times
10^6M_{\odot} (\upsilon_{\rm FWHM}/10^3{\rm km s^{-1}})^{6.67}$, called as the
Eddington limit. Second, the Eddington limit luminosity will be relaxed if the
trapped photons can escape from the magnetized super-Eddington accretion disk
via the photon bubble instability, and the size of the broad line region will
be enlarged according to the empirical reverberation relation, leading to a
relatively narrow width of H$\beta$. We call this the Begelman limit.
  Super-Eddington accretions in a sample composed of 164 AGNs have been
searched by this limit relation. We find there are a handful of objects locate
between the Eddington and Begelman limit lines, they may be candidates of
super-Eddington accretors in a hybrid structure of photon trapping and photon
bubble instability. The maximum width of H$\beta$ is in the reange of $(3.0 -
3.8)\times 10^3$ km s$^{-1}$ for the maximum mass black holes with
super-Eddington accretion rates among AGNs. We suggest that this limit relation
is more reliable and convenient to test whether a source is super-Eddington and
useful to probe the structure of the super-Eddington accretion process.
