Counting the minimal number of inflections of a plane curve

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 $diff(\gamma )$ may have, where $diff$ 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.