Holomorphic Mappings Preserving Minkowski Functionals

We show that the equality m1(f(x)) = m2(g(x)) for x in a neighborhood of a point a remains valid for all x provided that f and g are open holomorphic maps, f(a) = g(a) = 0 and m1, m2 are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f and g is obtained. Next we generalize these results to bounded quasi-balanced domains. Moreover, the main results of Berteloot and Patrizio ([Ber-Pat], 2009) and Boutat ([Bou], 2009) are significantly extended and their proofs are essentially simplified.