  Let $\Omega$ be a smooth oriented bounded domain in $\mathbb R^4$,
$H_0^2(\Omega)$ be the Sobolev space, and $\lambda_1(\Omega)= \inf \{\|\Delta
u\|_2^2 : u\in H_0^2(\Omega), \|u\|_2 =1\}$ be the first eigenvalue of the
bi-Laplacian operator $\Delta^2$ on $\Omega$. For $\alpha \in
[0,\lambda_1(\Omega))$, we define $\|u\|_{2,\alpha}^2 = \|\Delta u\|_2^2 -
\alpha \|u\|_2^2$, for $u \in H_0^2(\Omega)$. In this paper, we will prove the
following inequality \[ \sup_{u\in H_0^2(\Omega),\, \|u\|_{2,\alpha} \leq 1}
\int_{\Omega} e^{32 \pi^2 u(x)^2} dx < \infty. \] This strengthens a recent
result of Lu and Yang \cite{LuYang}. We also show that there exists a function
$u^*\in H_0^2(\Omega)\cap C^4(\overline{\Omega})$ such that $\|u^*\|_{2,\alpha}
=1$ and the supremum above is attained by $u^*$. Our proofs are based on the
blow-up analysis method.
