# Bound on the third derivative with Lipschitz condition

I'm trying to understand the following paragraph from Boyd & Vandenberghe, page 488:

(...) we assume that the Hessian of $$f$$ is Lipschitz continuous on $$S$$ with constant $$L$$, i.e., $$\| \nabla^{2}f(x) - \nabla^{2} f(y) \|_{2} \leq L \| x-y \|_{2}$$ for all $$x, y \in S$$. The coefficient $$L$$, which can be interpreted as a bound on the third derivative of $$f$$, can be taken to be zero for a quadratic function. More generally $$L$$ measures how well $$f$$ can be approximated by a quadratic model (...)

What exactly is the reason for stating a bound on the third derivative this way, rather than, say $$\| \nabla^{3} f(x) \|_{2} \leq M < \infty$$ for all $$x \in S$$? Are these two statements somehow identical, or does one imply the other? What is (if any) the relationship between $$L$$ and $$M$$ here?

Since the third derivative of the quadratic function is zero, I expect the reason for stating a bound on the third derivative of $$f$$ this way is simply to support an interpretation of $$L$$ as a measure how well $$f$$ can be approximated by a quadratic model, because when $$f$$ is already quadratic, $$L$$ can be taken to be zero.
I expect that the relationship between $$L$$ and $$M$$ here is that if $$\nabla^{3} f(x)$$ exists then we can take $$L=M$$. There is the following real analysis intuition for this. A differentiable function $$g:\Bbb R\to\Bbb R$$ is $$L$$-Lipschitz iff $$|g’(x)|\le L$$ for each $$x\in\Bbb R$$. Indeed, Implication $$(\Rightarrow)$$ follows from the definition of a derivative, Implication $$(\Leftarrow)$$ follows from Lagrange’s theorem, stating that for all real $$x there exists $$z\in (x,y)$$ such that $$g(y)-g(x)=g’(z)(y-x)$$.
In the case where the function is defined on all of $$\mathbb{R}^n$$, the implication from bounded third derivative to lipshitz hessian is exactly the mean value theorem for vector valued functions where we have that $$M =L$$.
For the direction you're asking about, consider $$y = x + h$$ for some small $$h$$ then take $$h \rightarrow 0$$. By definition of the norm of the gradient this shows bounded third derivative with $$M=L$$.