It is general known that if $X$ is a random variable, then for increasing functions $f,g$ we have that $Cov(f(X),g(X))\ge 0$.
I am trying to investigate a more specific result for joint,y distributed random variables. If $(X,Y)$ is a 2d normal distribution with $Cov(X,Y)\ge 0$, and if $f,g$ are now two different increasing functions does it necessarily hold that $Cov(f(X),g(Y))\ge 0$?