Let $f:\Omega \subseteq \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a smooth function. Then $$k_1(x-y)^T(x-y) \le (f(x)-f(y))^T(f(x)-f(y)) \le k_2(x-y)^T(x-y)$$ for any $x,y \in \Omega$, where $k_2>k_1$ are two arbitrary constants.($T$ denotes transpose operator).
I am wondering if this inequality has been used somewhere in the literature. Also I am not sure if this condition is only satisfied when $f(\cdot)$ is linear, i.e. f(x)=Ax with $A \in\mathbb{R}^{n\times n}$?
Provided that $0<k_1 \leq k_2$ (else you have triviality), $f$ satisfying this condition is bi-Lipschitz (equivalence of Wikipedia's definition and your inequality becomes apparent if you scale $f$). Such a map has to be continuous (obviously, by the upper bound), and the lower bound forces injectivity.