Hyperbolic metrics on universal Teichmüller space and extremal problems

The paper consists of two parts, both related to complex geometry of the universal Teichm¨uller space. We reprove that all contractible invariant metrics on this space coincide and apply this important fact to solving the general extremal problems for univalent functions with quasiconformal extensions

Extremal quasiconformality vs bounded rational approximation

We show that, on most of the hyperbolic simply connected domains, the weighty bounded rational approximation in a natural sup norm is possible only for a very sparse set of holomorphic functions (in contrast to the integral approximation). The obstructions are caused by the features of extremal quasiconformality.The paper is devoted to the 100th anniversary of Georgii Dmitrievich Suvorov, my first university adviser and teacher. He was an outstanding mathematician and a widely talented, extremely great human being