Is there any particular name for this condition: $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ such that $$L_1\Vert x-y \Vert\le\Vert f(x)-f(y)\Vert\le L_2\Vert x-y \Vert$$ for all different and nonzero vectors $x,y\in \mathbb{R}^n$, with $L_1\in\mathbb{R}$, $L_1<L_2$, and $L_2>0$
I know that the upper-bound is the Lipschitz condition, but what about the lower bound?
Any idea of which class of functions satisfies this inequality? I am not sure if a function $f(\cdot)$ with lower and upper bounded Jacobian satisfies this condition.