  There is clear evidence that the $\gamma$-ray emission from active galactic
nuclei (AGNs) is attributed to the inverse Compton scatterings in the
relativistic blobs near the massive black holes. If the soft seed photons are
from the regions outside the blobs, a linear relation between $(\nu
F_{\nu,\gamma}/\nu F_{\nu, \rm syn} u^{*})^{1/(1+\alpha)}$ and Doppler factor
$\delta$ is expected, where $\nu F_{\nu,\gamma}$ and $\nu F_{\nu, \rm syn}$ are
monochromatic $\gamma$-ray and synchrotron fluxes, respectively, and $u^{*}$ is
the energy density of soft seed photons \citep{D97}. We estimate the soft
photon energy density in the relativistic blobs contributed by the broad line
region (BLRs) in these $\gamma$-ray AGNs using their broad-line emission data.
The Doppler factors $\delta$ are derived from their radio core and X-ray
emission data, based on the assumption that the X-ray emission is produced
through synchrotron self-Compton (SSC) scatterings. We find two nearly linear
correlations: $(\nu F_{\nu,\gamma}/\nu F_{\rm opt} u^{*})^{1/(1+\alpha)}
\propto \delta^{1.09}$, and $(\nu F_{\nu,\gamma}/\nu F_{\rm IR}
u^{*})^{1/(1+\alpha)} \propto \delta^{1.20}$, which are roughly consistent with
the linear correlation predicted by the theoretical model for external Compton
scatterings. Our results imply that the soft seed photons are dominantly from
the BLRs in these $\gamma$-ray AGNs.
