Harmonic Convolution And Harmonic Extension

Harmonic mapping is a natural generalization of conformal mappings.The theory of harmonic mappings initially connects with the theory of min-imal surfaces,the complex analysis scholars Clunieand Sheil-Small later point that many classical conclusions of conformal mappings can generalize to harmonic mappings,and make it has the similar conclusions.This paper mainly studies the correlation properties of a subset of harmonic functions and the harmonic quasi conformal extension of boundary functions.1.A class of Salagean type single leaf harmonic functions is studied,and the quasi conformal,convexity and convolution properties of this kind of functions are obtained,and the related results are improved.2.A class of increasing self homeomorphism and its convex combination on the real axis as boundary functions,and estimate their dilatation functions.This paper studies the extension to the upper half plane harmonic quasicon-formal homeomorphism,estimate its dilatation function and the dilatation function.