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.
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