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For any finitely generated group $G$, two complexity functions $\alpha_G$ and$\beta_G$ are defined to measure the maximal possible gap between the norm ofan automorphism (respectively outer automorphism) of $G$ and the norm of itsinverse. Restricting attention to free groups $F_r$, the exact asymptoticbehaviour of $\alpha_2$ and $\beta_2$ is computed. For rank $r\geqslant 3$,polynomial lower bounds are provided for $\alpha_r$ and $\beta_r$, and theexistence of a polynomial upper bound is proved for $\beta_r$.

Bounding the gap between a free group (outer) automorphism and its inverse