Consider a function $f:\mathbb{R}^d_+ \to \mathbb{R}^d_+$ such that $f(0) = 0$. Is there a name for functions that satisfy the following condition
$$\frac{||f(x)-f(y)||}{||f(x)||} \le \frac{||x-y||}{||x||} $$
or in other words, they do not expand the relative error. The condition can be somewhat massaged to necessary condition:
$$\frac{||f(x)||}{||x||} \ge ||\nabla f(x)|| $$
Note that if $d=1$ then the condition implies concavity of $f$ and vise-versa. However, this does not seem to be the case when $d>1$ so was wondering if there is any specific term for this.