On the extremality of quasiconformal mappings and quasiconformal deformations

Abstract

Abstract. Given a family of quasiconformal deformations F (w, t) such that ∂F has a uniform bound M , the solution f(z, t)(f (z, 0) = z) of the Löwner-type differential equationis an e 2Mt -quasiconformal mapping. An open question is to determine, for each fixed t > 0, whether the extremality of f (z, t) is equivalent to that of F (w, t). The note gives this a negative approach in both directions.