Integral Inequality with L-2 Norm

On page 135 of The Mathematical Theory of Finite Element Methods (Brenner and Scott), I encountered the following inequality:

$$\left | \int_{\Gamma} \overline{v} - v \, ds \right | \leq |\Gamma |^{1/2} || \overline{v} - v ||_{L^2(\Gamma)}\,$$.

The mean $$\overline{v}$$ is defined as

$$\overline{v} = \frac{1}{\text{meas}(\Omega)} \int_{\Omega} v(x) \, dx \,$$

and $$\Gamma = \partial \Omega$$.

The inequality is part of a proof that the bilinear form for the Poisson equation with Dirichlet boundary conditions is coercive.

My question is: could I replace the $$\overline{v}-v$$ on both sides of the inequality with some (more general) function? If so, what would the restrictions on the function be? I'm not sure where the inequality comes from.

This is just Hoelder inequality. Let me denote $$w:=\bar v -v$$. Then $$|\int_\Gamma w \ ds| \le \int_\Gamma |w|\cdot 1 \ ds \le \|1\|_{L^2(\Gamma)} \|w\|_{L^2(\Gamma)} = |\Gamma|^{1/2}\|w\|_{L^2(\Gamma)}.$$ This works if $$w\in L^2(\Gamma)$$ and $$|\Gamma|<+\infty$$.