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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. As a con...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2014-01, Vol.409 (2), p.643-648
Main Author: Kosiński, Łukasz
Format: Article
Language:English
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Summary: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. 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.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2013.07.030