  Given a plane curve $\gamma: S^1\to \mathbb R^2$, we consider the problem of
determining the minimal number $I(\gamma)$ of inflections which curves
$\mbox{diff}(\gamma)$ may have, where $\mbox{diff}$ runs over the group of
diffeomorphisms of $\mathbb R^2$. We show that if $\gamma$ is an immersed curve
with $D(\gamma)$ double points and no other singularities, then $I(\gamma)\leq
2D(\gamma)$. In fact, we prove the latter result for the so-called plane
doodles which are finite collections of closed immersed plane curves whose only
singularities are double points.
