  In this paper, we establish a Modica type estimate on bounded solutions to
the overdetermined elliptic problem \begin{equation*}
  \begin{cases}
  \Delta u+f(u) =0& \mbox{in $\Omega$, }\\ u>0 &\mbox{in $\Omega$, }
  u=0 &\mbox{on $\partial\Omega$, }
  \partial_{\nu} u=c_0 &\mbox{on $\partial\Omega$, }
  \end{cases} \end{equation*} where $\Omega\subset\mathbb{R}^{n},n\geq 2$. As
we will see, the presence of the boundary changes the usual form of the Modica
estimate for entire solutions. We will also discuss the equality case. From
such estimates we will deduce information about the curvature of $\partial
\Omega$ under a certain condition on $c_0$ and $f$. The proof uses the maximum
principle together with scaling arguments and a careful passage to the limit in
the arguments by contradiction.
