  We prove that the following pointwise inequality holds
  \begin{equation*} -\Delta u \ge \sqrt\frac{2}{(p+1)-c_n} |x|^{\frac{a}{2}}
u^{\frac{p+1}{2}} + \frac{2}{n-4} \frac{|\nabla u|^2}{u} \ \ \text{in}\ \
\mathbb{R}^n
  \end{equation*}
  where $c_n:=\frac{8}{n(n-4)}$, for positive bounded solutions of the fourth
order H\'{e}non equation that is \begin{equation*} \Delta^2 u = |x|^a u^p \ \ \
\ \text {in }\ \ \mathbb{R}^n \end{equation*} for some $a\ge0$ and $p>1$.
Motivated by the Moser's proof of the Harnack's inequality as well as Moser
iteration type arguments in the regularity theory, we develop an iteration
argument to prove the above pointwise inequality. As far as we know this is the
first time that such an argument is applied towards constructing pointwise
inequalities for partial differential equations. An interesting point is that
the coefficient $\frac{2}{n-4}$ also appears in the fourth order $Q$-curvature
and the Paneitz operator. This in particular implies that the scalar curvature
of the conformal metric with conformal factor $u^\frac{4}{n-4}$ is positive.
