My analysis professor said that if $V \colon \mathbb R^n \to \mathbb R$ is $\mathcal C^{1,\alpha}$ for $\alpha >0$, i.e. it is differentiable with Holder's continuous gradient of order $\alpha$ then we have: $$\left|V\left(y_1\right)-V\left(y_0\right)\right| \leq C\left|y_1-y_0\right|^{1+\alpha}.$$ I don't have any idea how to prove it but it doesn't sound true to me as for example $f(x)=x^2$ is $\mathcal C^{1,1}$ but $f$ is not even Lipschitz.
Myabe I missed something, is that property true at least on bounded subsets of $\mathbb R^n$ and if yes, how is it proved?
