Let $G:\mathbb R\to\mathbb R$ be a concave function, define $G_{\epsilon}: \mathbb R\to\mathbb R$ by
$$G_{\epsilon}(x)~~:=~~\max_{y\in [x-\epsilon, x+\epsilon]}G(y).$$
My question is the following: Is there some $h:\mathbb R\to \mathbb R$ s.t.
$$G_{\epsilon}(x)~+~h(x)(y-x)~-~\epsilon|h(x)|~~ \ge~~ G(y)\mbox{ for all } x, y \in\mathbb R?$$
PS: This question is related to the question of the following link