Holomorphic mappings preserving Minkowski functionals

We show that the equality m 1 ( f ( x ) ) = m 2 ( 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 m 1 , m 2 are Minkowski functionals of bounded balanced domains. Moreover, a polynomial relation between f and g is obtained. onsequence of our considerations we extend the main result of Berteloot and Patrizio (2000) [2] and we simplify its proof. o show how to apply our results to quasi-balanced domains.