Upper bounding Lipschitz constant of differentiable function

Question

Let's assume we have a k-differentiable function with an unknown Lipschitz constant, $f:\mathbb{R}^m \rightarrow \mathbb{R}^n$.

How can we upper bound $Lip(f)$? Can we somehow use that Jacobian or Hessian to bound it?

Answer 1

Yes : on a convex set $C$, you know that if the norm of the Jacobian (seen as a linear application) is bounded by $k$ on this convex set, then, for all $x,y\in C$ $$ \|f(a)-f(b)\|\leq k\|a-b\|$$ If $C$ is the whole space, you have an upper bound for Lip($f$).