  The sizes of the broad-line regions (BLRs) in $\gamma$-ray active galactic
nuclei (AGNs) are estimated from their optical continuum luminosity by using
the empirical relation between $R_{\rm BLR}$ and $L_{\lambda,\rm opt}$. Using
the broad emission line data, we derive the photon energy density in the
relativistic blobs near the massive black holes in AGNs. We calculate the power
of the broad-line photons Compton up-scattered by the relativistic electrons in
the blobs. Compared with observed $\gamma$-ray emission data, the Doppler
factors $\delta$ of the blobs for a sample of 36 $\gamma$-ray AGNs are derived,
which are in the range of $\sim 3-17$. We estimate the central black hole
masses of these $\gamma$-ray AGNs from the sizes of the BLR and the broad line
widths. It is found that the black hole masses are in the range of $\sim
10^{8}-10^{10}M_{\odot}$. A significant correlation is found between the
Doppler factor $\delta$ and the core dominance parameter $R$. The results are
consistent with the external radiation Compton (ERC) models for $\gamma$-ray
emission from AGNs. The soft seed photons are probably from the broad-line
regions.
