Uniform lower bound of the inner product of two positively correlated random variables

Question

Suppose $W$ is a mean-zero random variable with unit variance, i.e., $E(W)=0,Var(W)=1$. Let $g(\cdot)$ be a non-constant increacing function such that $g(W)$ has zero mean and variance $\epsilon>0$, i.e., $E(g(W))=0,Var(g(W))\ge\epsilon$. Can we have a non-zero lower bound on $E\{g(W)W\}$ uniformly for all such $g$? Or equivalently, does there exist some positive constant $c>0$ such that $E\{g(W)W\}>c$ hold for all such $g$?