Isoperimetric growth theorems for quasiconformal automorphisms of a circle (Q1390413)

The author considers the class \(Q_k\) of \(K\)-quasiconformal automorphisms of the open unit disc \(U\) satisfying \(f(0)=0\), \(f(1)=1\) and studies the problem of values of \(| b(r_2)|\) for given \(| f(r_1) |\) where \(0\leq r_1<r_2<1\). The method is the method of modules of multiple curve families originated by \textit{J. A. Jenkins} (Ann. Math. 66, 440-453 (1957; Zbl 0082.06001)]; Tôhoku Math. J. 45, No. 2, 249-257 (1993; Zbl 0780.30019)]. By making several choices of free families of homotopy classes he defines corresponding extremal mappings in terms of which he describes the solution of his problem. He also discussion the boundary of the set of values \((| f(r_1) |\), \(| f(r_2) |)\) which corresponds to \(\max_{f\in Q_k} | f(r_2) |\) for given \(| f(r_1) |\).