Boundary correspondence, principle of

A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping if the facts that is a continuous mapping of the closure of a domain onto the closure of a domain and is a homeomorphism of onto imply that is a topological mapping of onto . Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. Boundary correspondence (under conformal mapping)).

If and are plane domains with Euclidean boundaries homeomorphic to a circle and is bounded, then the principle of boundary correspondence holds for analytic functions on , i.e. is a conformal mapping of onto . In addition to that given above, various other forms of the principle of boundary correspondence are commonly used for conformal mappings (see [1]). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see [2]).

References

[1]	M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian)
[2]	L.D. Kudryavtsev, "On differentiable mappings" Dokl. Akad. Nauk SSSR , 95 : 5 (1954) pp. 921–923 (In Russian)
[3]	S.I. Pinchuk, "Holomorphic equivalence of certain classes of domains in " Math. USSR-Sb. , 111 (153) : 1 (180) pp. 67–94; 159 Mat. Sb. , 111 (153) : 1 (1980) pp. 67–94; 159