A new rigidity result for holomorphic maps

In this paper we determine which vanishing order of a holomorphic map f at a point of the (non necessarily regular) boundary of a very generic domain of is required for f to be constant. In particular this vanishing order is 1 if the boundary is Dini-smooth whereas it is at least β/α if f locally maps a Dini-smooth corner of opening πα into a Dini-smooth corner of opening πβ. Finally an analogous result is stated for the case of a holomorphic map f which maps a cusp into a cusp.