Lipschitz continuity result

Question

Hi: I'm reading a proof in a convex optimization book and I don't understand something. The author states the following:

Let $Q$ be a subset of $R^{n}$. We denote by $C_{L}^{k,p}(Q)$, the class of functions with the following properties:

A) any $f \in C_{L}^{k,p}(Q)$ is $k$ times differentiable on $Q$.

B) Its $p$-th derivative is Lipschitz continuous on $Q$ with the constant $L$.

i.e: $\|f^{p}(x) - f^{p}(y)\| \le L\|x - y\|$.

This is fine. But then later, inside the proof, he states the following:

if $f \in C_{L}^{2,1}(R^{n})$, then for any $s \in R^{n}$ and $\alpha > 0$, we have

$$\left\|\left(\int_{0}^{\alpha}f''(x + \tau ~ s)d ~ \tau\right) s\right\| = \|f^{\prime}(x + \alpha s) - f^{\prime}(x) \| \le \alpha L \|s\|\;.$$

I don't understand how this is implied by the definition of Lipschitz continuity defined previously. Thanks a lot.

Answer 1

Just putting in the definition of $f\in C^{2,1}_L(\mathbb{R}^n)$ gives

$$\|f^{(1)}(x+\alpha s)-f^{(1)}(x)\|\le L\|x+\alpha s -x\|=L\|\alpha s\| =L\alpha \|s\|$$

where I used positive homogeneity of the norm $\|\cdot\|$ in the last equality.