Estimating curve length in terms of the curvature Let $\Gamma$ be a $C^2$ curve in $[0,1]^2$ connecting two points $\mathbf{p}$ and $\mathbf{q}$. Let $\ell$ and $K$ be the length of  $\Gamma$ and the maximum curvature of $\Gamma$, respectively.  Is there an inequality that bounds $\ell$ from above by a function of $K,\mathbf{p}$ and $\mathbf{q}$?

Definitions of $\ell$ and $K$: Let $\gamma:[a,b]\to [0,1]^2$ be an arc-length parametrization ($|\gamma'(t)|=1$ for all $t\in[a,b]$) such that $(\mathbf{p},\mathbf{q})=(\gamma(a),\gamma(b))$, then we define $\ell$ and $K$ as  $$\ell=b-a,\qquad ,\qquad K=\max_{a\le t<b} |\gamma''(t)|. $$
We know that $\ell\ge |\mathbf{p}-\mathbf{q}|$, is there a bound on $\ell$ from above as well like $\ell\le |\mathbf{p}-\mathbf{q}|+C K$?
 A: Maybe you are looking for Schur's Theorem for convex curves:

Given are two curves $\alpha$, $\gamma$ with positive curvature, both of length $\ell$. If $|k_\alpha(s)| \geq |k_\gamma(s)|$, $k_\alpha(s) \neq k_\gamma(s)$, the chord subtended by $\gamma$ is bigger than that subtended by $\alpha$.

The theorem is very intuitive. Take a piece of wire and lay it on a table. The harder you curve the wire, the closer its endpoints will lie.
Let $K$ be the maximum curvature of $\gamma$, and let $\alpha$ be the circle segment with radius $1/K$. By Schur's Theorem
$$
 \|\gamma(b) - \gamma(a)\| \geq \|\alpha(b) - \alpha(b)\|.
$$
If the length is not too long, the length of the chord of the circle is $2R \sin(\frac{\theta}{2})$. Here $\theta$ is the angle of the circle segment and $0 < \theta \leq \pi$. Since the circle segment has length $\ell$, we have $\ell = R\theta = \frac{\theta}{K}$. Hence
$$
  \|\gamma(b) - \gamma(a)\| \geq 2R \sin\left( \frac{\ell}{2R}\right) = \frac{2}{K} \sin \left(\frac{K\ell}{2}\right).
$$
In particular, if $\gamma$ has length $\ell = R\pi$, then its end points lie at least $2R$ away from each other.
