  We consider a conformal field theory in the presence of a boundary, and
explain how two-point correlators of mixed bulk-local operators can be
bootstrapped by exploiting the analytical structure of the conformal blocks.
This yields the operator product expansion coefficients in either bootstrap
channel. We apply this bootstrap technique to an $O(N)$-model in $d = 4 -
\epsilon$ dimensions to find the $\phi - \phi^5$ bulk correlator and the
corresponding operator product expansion coefficients upto
$\mathcal{O}(\epsilon)$.
  Parts of this paper was first presented in my thesis.
