Some Problems On Quasiconformal Mappings

The theory of quasiconformal mappings in the complex plane is well developed and plays important roles in study of Teichmuller spaces, Riemann surfaces, Fuchian group and complex dynamic systems, etc. Suppose that / is a quasiconformal mapping with complexdilatation (z), . Many mathematicians have been interested in the problems of the existence of (z)-homeomorphisms solutions of the Beltrami equation when = 1. In section 2 of this dissertatiom, we give an estimate on h(x,t) under Brakalova-Jenkins'conditions.The research on the inner radius of univalence attract many people's attention. Galvis , Lehto , Lehtinen, Miller - VanWieren obtained many important results. The inner radius of univalence for some particular domains such as triangles, regular polygons and angle sectors have been obtained. Part estimations of inner radius of univalence for ellipses and rectangles have also been obtained. In section 3 of dissertation, we obtain the inner radius of univalencefor a class of inequiangular hexagons from classic Schwarz -Christoffel formula.Extremal mappings have been one of the main topics in the theory of quasiconformal mappings, n the section 4, we consider the extremal mappings on the surface Rwhere every Ri is a hyperbolic Riemann surface, Ri Rj = ,i j,I is a non-empty indexset. Some results on extremal mappings in the classical quasiconformal mappings are extended to this surface R.