The decomposition theorem for the family of complete quasi-convex mappings
The authors recall Suffridge's decomposition theorem according to which a normalized locally biholomorphic mapping \(f(z)=(f_1(z),f_2(z),\dots,f_n(z))\) on the unit polydisk \(\Delta^n\) in \(\mathbb C^n\) is a normalized biholomorphic convex mapping of \(\Delta^n\) if and only if \(f(z)=(g_1(z_1),g_2(z_2),\dots,g_n(z_n))\) where \(g_1,g_2,\dots,g_n\) are biholomorphic convex functions on \(\Delta\) in \(\mathbb C\), and \(z=(z_1,z_2,\dots,z_n)\in\Delta^n\). Taishun Liu and Guangbin Ren extended this result from \(\Delta^n\) to the direct product of a finite number of bounded convex circular domains. The main result of the paper generalizes the decomposition theorem for a more general family of mappings. Namely, the authors showed that, for the direct product of a finite number of bounded convex circular domains, the decomposition theorem holds not only for the family of convex mappings, but also for the family of complete quasi-convex mappings of type \((n_1,n_2,\dots,n_k)\) introduced in the article. Thus a family of convex mappings is decomposed as a direct product of families of convex mappings while a family of complete quasi-convex mappings of type \((n_1,n_2,\dots,n_k)\) is decomposed as a direct product of families of quasi-convex mappings.
complete quasi-convex mapping of type \((n_1
scientific article; zbMATH DE number 2126900