In the second chapter of Spivak's Calculus in Manifolds the following theorem is proved:
Theorem: Let $A\subseteq \mathbb{R}^n$ be a rectangle and let $f:A\to \mathbb{R}^n$ be $C^1$. If there is a number $M$ such that $|D_jf_i(x)|\le M$ for all $x$ in the interior of $A$, then $$|f(x)-f(y)|\le n^2M|x-y|$$ for all $x,y\in A$.
I was wondering if there is a similar theorem where $|f(y)-f(x)|$ is in the "larger than" part of the inequality.
Letting $$m=\inf |\partial_jf_i(x)|$$ then assuming $(y_j-x_j)\ge 0$ and $\partial_jf_i\ge 0$ in $A$, the Mean Value Theorem (in a single variable) gives
$\begin{align} |f(y)-f(x)|^2 & =\sum_i\left(\sum_j(y_j-x_j)\partial_jf_i(z_{ij})\right)^2\\ & \ge m^2 \sum_i\left(\sum_j(y_j-x_j)\right)^2\\ & \ge m^2 \sum_i\left(\sum_j(y_j-x_j)^2\right)\\ & \ge m^2 \sum_i|y-x|^2\\ & = nm^2|y-x|^2.\\ \end{align}$
Therefore $$|f(y)-f(x)|\ge \sqrt{n}m|y-x|.$$