  Let $AL^{2}_{\phi}(\mathbb{D})$ denote the closed subspace of
$L^{2}(\mathbb{D},e^{-2\phi}d\lambda)$ consisting of holomorphic functions in
the unit disc ${\mathbb D}$. For certain class of subharmonic funcions $\phi :
{\mathbb D}\rightarrow{\mathbb D}$, we prove upper pointwise estimate for the
Bergman kernel for $AL^{2}_{\phi}(\mathbb{D})$
