Given a free group $F_r$ on $A = \{ a_1, \ldots, a_r\}$ and an automorphism $\varphi$ of $F_r$, we can consider the norm $||\varphi|| = |a_1\varphi| + \ldots + |a_r\varphi|$. How big can be $||\varphi^{-1}||$ relatively to $||\varphi||$? More precisely, this talk concerns the complexity of the function $n \mapsto \max\{ ||\varphi^{-1}|| : ||\varphi|| \leq n \}$. We claim that this complexity is at least $O(n^r)$ and precisely quadratic if $r = 2$. Similar results hold if we consider the other natural norms on the automorphism group. These results were obtained in joint work with Manuel Ladra and Enric Ventura.