Some Problems On Quasiconformal Mappings And The Inner Radius Of Univalency

In the first part of this article, we discuss the problem on the inner radius of uni-valency for an equiangular hexagon H whose sides form the sequence baabaa. L.Wieren proved that if 1 < b/a < 1.67117..., then H is a Nehari disk and a(H) = 8/9 = (P6). Using the methods developed by L.Wieren, we prove that H is still a Nehari disk and a(H) = 8/9 = (-Pe) when 0.6157... < b/a < 1. In the proof, we use the Mathematica software package and the Maple software package.In the second part of this article, we extend some results on extremal quasiconfor-mal mappings between hyperbolic Riemann surfaces. On the surface R = Ri, whereevery Ri is a hyperbolic Riemann surface, Ri Rj = , i j, and / is a non-empty index set, we introduce the concepts of extremality, unique extremality, infinitesimal extremality and unique infinitesimal extremality, and extend the Hamilton-Krushkal condition for extremal Beltrami differentials of the hyperbolic Riemann surface to this case. We also discuss the extremal problem for Teichmuller spaces of closed subsets of the sphere and obtain some similar results.