Local monotonicity of measures supported by graphs of convex functions

Let $f \in C^2(\mathbb{R})$ satisfy $f(0)=f'(0)=0$ and $f''(0) > 0$. Then the $1$-dimensional Hausdorff measure restricted to the graph of $f$ is locally monotone near the origin in the sense that there exists $\sigma>0$ such that the function $r \mapsto \frac{\mu_fB(z,r)}{r}$ is nondecreasing on $(0,\sigma)$ for every centre $z \in B(\sigma)$.

The result is reformulated for Hausdorff measures restricted to uniformly $C^2$-curves in $\mathbb{R}^2$ with the curvature bounded away from zero and infinity.