Module inequalities for quasiregular mappings
Cabiria Andreian Cazacu · Annales Academiae Scientiarum Fennicae Series A I Mathematica · 1976
CAZACUThe problem of finding the z-dimensional generalization of the theory of complex analytic functions in the plane gave rise to different important research fields among which the relatively recent theory of quasiregular mappings is one of the most successful.This theory has its roots in the theory of quasiconformal mappings (i.e.quasiconformal homeomorphisms) and in Stoilow's topological theory of analytic functions.It was first developed.by Ju.G. Reöetnjak t8l -[0] and by O. Martio, S. Rickman and J. Våisälö t2l -t4l.As in quasiconformality, the module of curve or surface families con- stitutes a useful instrument in the study of quasiregularity.Several authors, O. Martio, S. Rickman and J. Våisälå [2], E. A. Poleckii [6], J. Våisålå [11], and others, obtained different generalizations of Grötzsch module ine- qualities from 2-and zl-dimensional quasiconformality to quasiregularrty.These results permitted one to characterize quasiregular mappings in terms of modules and to establish numerous properties for them.The purpose of our pa,per is to prove other module inequalities which improve the above mentioned ones in the following respscfu: l.Using the nodule with weight we need not use global dilatations, and the deduced bounds are sharper.In this way we obtain also equality cases.Tho results are given for the module with order, too.2. The inequalities and equalities we prove are valid not only for curves Univorsity of Bucharost