I got the following inequality:

$B_{4}$ is the unit ball in $R^{4}$, $\partial B_{4}$ is its boundary.

$(\int_{B_{4}}e^{4u}dx)^{\frac{1}{4}} \leq S(\int_{\partial B_{4}}e^{3u}d\xi)^{\frac{1}{3}}$,

for all $u$ satisfies $\Delta^{2}u\leq 0$, $-\frac{\partial u}{\partial\gamma}\leq 1$, $\gamma$ is the outer nomal of $\partial B_{4}$.

$S$ is the sharp constant which can be determined by letting $u$ solve $\Delta^{2}u=0$,

$-\frac{\partial u}{\partial\gamma}=1$, $u=0$ on $\partial B_{4}$.

The above inequality is also conformally invariant in the sense that it is invariant if one replace $u$ by the function $u\circ \tau+\frac{1}{4}ln(J_{\tau})$, where $\tau$ is the conformal map from $B_{4}$ to itself, $J_{\tau}$ is its Jacobian.

Does anyone see similar inequality in the literature? What is the criteria for an inequality to be good?

Any comments or refference will be appreciated..