  We consider the behavior of the quantity $p(\beta)$; the free energy of
directed polymers in random environment in $1+2$ dimension, where $\beta$ is
inverse temperature. We know that the free energy is strictly negative when
$\beta$ is not zero. In this paper, we will prove that $p(\beta)$ is bounded
from above by $-\exp\left( -\frac{c_\varepsilon}{\beta^{2+\varepsilon}}
\right)$ for small $\beta$, where $c_\varepsilon>0$ is a constant depending on
$\varepsilon>0$. Also, we will suggest a strategy to get a sharper asymptotics.
