Hey guys I need your help.
Let $\Omega$ be a bounded, 2 or 3 dimensional domain with smooth boundary. Let $c\in H^2(\Omega)$ with Neumann boundary conditions. We define $\overline{c}=\frac{1}{|\Omega|}\int_{\Omega} c dx$ and $f\in C^\infty(\mathbb{R},\mathbb{R})$ be convex.
How do I get an approximation like
$f'(c)(c-\overline{c})\geq C_1|f'(c)|-C_2$?
I really have no clue. I think we were ok if instead of $c$ we take $x$ for $f(x)$ and $\overline{x}\in\mathbb{R}$. There don't has to be a $C_2$ but it can ;).