Parabolic Starlike Mappings In Several Complex Variables

In this paper, we investgated: parabolic starlike mappings,parabolic starlike map-pings of orderρ,parabolic and spiralike mappings of typeβ,parabolic and spiralikemappings of typeβand orderρ. Among which parabolic starlike mappings can be inte-grated in the parabolic starlike mappings of orderρ, parabolic and spiralike mappings oftypeβcan be integrated in parabolic and spiralike mappings of typeβand orderρ, andthe former three types of mapping can be unified in parabolic and spiralike mappings oftypeβand orderρ. Thus we can only study parabolic and spiralike mappings of typeβand orderρ, the former three mappings's results can be regarded as the corollary of itself.In this article, we mainly research: the various of Roper-Suffridge extension opera-tors maintain that the nature of the former four types of mapping classes which we newlydefined is invariable, that is to say, from the Roper-Suffridge extension operator,we canobtain the corresponding mappings classes in high dimensional space by the functions inthe unit disc. This nature offer us the possibility to structure examples in high dimen-sion.From these concrete examples, we can also require ideas and methods to study themappings'other nature.All of these results were not known previously, the conclusion we got enrich thecontent of geometric function theory in several complex variables.