Suppose there is a function $f$, for which we know the inequality
$$f(r)\leq r$$ is true, where $r=||x-y||_2=\sqrt{(x-y)^T (x-y)}$ is the Euclidean distance. If now we use the Mahalanobis distance $r_M=\sqrt{(x-y)^T M^{-1} (x-y)}$, where M is a symmetric positive definite matrix, is the relation
$$f(r_M)\leq r_M$$ still true in general?