How to prove Poincare-like inequality?

Question

Suppose $u\in W^{1,1}$ and $\partial u$ is $C^1$. I want to prove the following: $\int_{\partial\Omega}|u-\bar u|\leq A\int_{\Omega}|\nabla u|$, where $\bar u=\dfrac{1}{|\Omega|}\int_{\Omega}u$ and $A>0$. Note that unlike Poincare inequality, the left integral is over $\partial\Omega$ and not $\Omega$. How can I do that?