Is there a generalization of the following statement for Holder continuous functions: For $f:\mathbb{R}^d \rightarrow \mathbb{R}$ Lipschitz continuous, $\|\nabla f\|$ is uniformly bounded.
What I am looking for is something along the lines: If $f$ is Holder continous with exponent $\alpha$, $$\|\nabla f(x)\| \leq C f(x)^{2-2\alpha}.$$
If the above statement was correct (it is not), for $\alpha=1$, we recover the previous statement for the Lipschitz functions.