Is this Poincaré-type inequality valid?

It is proved that the Poincaré inequality is still true for functions with zero mean boundary traces. Motivated by this, I have the following question:

Let $\Omega$ be an open,bounded and connected subset of $\mathbb R^3$ with a $C^2-$boundary $\partial \Omega \equiv \Gamma$. If $f \in W^{1,2}(\Omega)$ then could we claim that:

${\vert \vert f - \frac{1}{\vert \Gamma \vert } \int_{\Gamma} {\vert f \vert}^{1/2} \vert \vert}_{L^2(\Omega)} \leq C {\vert \vert \nabla f \vert \vert}_{L^2(\Omega)}$

If for any $u\in \{ u\in H^1(\Omega): \frac{1}{\vert \Gamma \vert } \int_{\Gamma} u=0 \}$ we have the estimate: ${\vert \vert u \vert \vert}_{L^2(\Omega)} \leq C {\vert \vert \nabla u \vert\vert}_{L^2(\Omega)}$ then it seems logical to me that the above claim could be true. However I wasn't able to prove it (if indeed can be proved) so any help is much appreciated.

Thanks in advance!

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