Quasiconformal Harmonic Functions Between Convex Domains

David Kalaj

We generalize Martio's paper [14]. Indeed the problem studied in this paper is under which conditions on a homeomorphism $f$ between the unit circle $S^1:=\{z:|z|=1\}$ and a fix convex Jordan curve $\gamma$ the harmonic extension of $f$ is a quasiconformal mapping. In addition, we give some results for some classes of harmonic diffeomorphisms. Further, we give some results concerning harmonic quasiconformal mappings (which follow by the results obtained in [10]). Finally, we give some examples which explain that the classes defined in [14] are not big enough to enclose all harmonic quasiconformal mappings of the disc onto itself.