Convolutions Of Harmonic Mappings And Quasiregular Extensions Of Meromorphic Functions

As a kind of generalization of analytic functions,harmonic mappings are closed related to univalent functions,quasiconformal mappings theory and Teichmüller space.The study of harmonic mappings enriches the theory of complex analysis.In these years,many scholars are concerned on convolution,univalence criterion and quasiconformal extensions for harmonic mappings.In this paper,we will continue to study these problems.Main results are as follows:Part one,we study univalence criterion for convolutions of plane harmonic mappings.It is shown that if the second dilation of a harmonic mapping f?z? in the right half-plane is Blascke product of second order,then its convolution with a canonical mappings f₀?z? of the right half plane is both univalent and sense-preserving.Furthermore,it is also convex in direction of the real axis.This result generalizes the former result,the second dilation of a harmonic mapping f?z? in the right half-plane is Blascke product of one order or a_n=0 in the literature[4-7].Part two,we study k-quasiregular extensions for meromorphic functions of a nonzero m-order pole.First,a sufficient condition of k-quasiregular extension for the class of meromorphic functions ??? is given.Next,we obtain an area theorem for the class functions of meromorphic functions ???.Last,we study the convolution of two meromorphic functions in ???.The result generalizes the cases of meromorphic functions of a nonzero first-order pole or zero pole in the literature[11-13].