Increase in curvature over a short distance

Question

Let $\Sigma$ be a smooth surface in $\mathbb{R}^3$. Let $p, q \in \Sigma$ such that $d_{\mathbb{R}^3}(p,q) < \epsilon$. Suppose that $\epsilon$ is sufficiently small so that the ball $B$ centered at $p$ for radius $\epsilon$ intersects $\Sigma$ in a single connected component, a $2$-ball. Thus the distance between $p$ and $q$ on the manifold $d_{\Sigma}(p,q) \leq d_{\mathbb{R}^{3}}(p,q) < \epsilon$. Let $\kappa$ be the maximum principal curvature at $p$ in absolute value. The maximum principal curvature varies smoothly as we walk along $\Sigma$ from $p$ to $q$ within $B$. How much larger can the maximum principal curvature at $q$ be, than at $p$, as a function of $\epsilon$?