On Decomposition Theorem of normalized biholomorphic convex mappings in Reinhardt domains

摘要

The construction of normalized biholomorphic convex mappings in the Reinhardt domain D_p = {(z_1,z_2,…,z_n) ∈ C~n : |z_1|~(p_1) + |z_2|~(p_2) + … + |z_n|~(p_n) < 1}, (p_j > 2, ,j = 1,2, … n) of C~n is discussed. The authors set up a Decomposition Theorem for such mappings. As a special case, it is proved that, for each such mapping f, the first k-terms of the homogeneous expansion of its j-th component f_j, j = 1, 2, … , n, depends only on z_j, where k is the number that satisfies k < min{p_1,p_2, … ,p_n} ≤ k + 1. When p_1,p_2, … ,p_n → ∞, this derives the Decomposition Theorem of normalized biholomorphic convex mappings in the polydisc which was gotten by T.J. Suffridge in 1970.