Holomorphic mappings preserving Minkowski functionals

We show that the equality m1(f(x))=m2(g(x))m1(f(x))=m2(g(x)) for xx in a neighborhood of a point aa remains valid for all xx provided that ff and gg are open holomorphic maps, f(a)=g(a)=0f(a)=g(a)=0 and m1,m2m1,m2 are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between ff and gg is obtained.As a consequence of our considerations we extend the main result of Berteloot and Patrizio (2000) [2] and we simplify its proof.We also show how to apply our results to quasi-balanced domains.