Given two random variables (say standard Normals) that are not necessarily independent, are there non-linear functions for which $$ \text{Cov}(g(X), g(Y)) \le c(g) \text{Cov}(X,Y), $$ where $c(g)$ is some constant that can depend on the function $g$. In other words, are there classes of functions $g$ that have covariance not too different from the covariance of the underlying variables?
update: Given the comments, my question is ill-posed, what I would like is an analog of Lipschitz continuity, so it should read that for any two gaussian random variables, the above bound holds, so the function $g$ can alter the covariance of the original variables by at most a fixed constant.