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

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 of an automorphism (respectively, outer automorphism) of G and the norm of its inverse. Restricting attention to free groups F-r, the exact asymptotic behaviour of alpha(2) and beta(2) is computed. For rank r >= 3, polynomial lower bounds are provided for alpha(r) and beta(r), and the existence of a polynomial upper bound is proved for beta(r)