Let $f:\mathbb{R^d}\to \mathbb{R}$ be a convex function. Let $<\cdot,\cdot>$ denote the usual Euclidean inner product. What can we say about $$ <x,\nabla f(x)>~~? $$
Is it true that $\exists$ $\eta,m >0$ such that for $|x|>m$
$$ <x,\nabla f(x)>\geq \eta |x|^2 $$
i.e there is some sort of dissipativity. What if $f$ was strictly convex?