On the extremal quasiconformal reflection. (Q2769245)

In numerous papers the author has considered the question of quasiconformal reflection. For a set \(E\) on the sphere a quasiconformal reflection is a sense-reversing self-homeomorphism of the sphere with bounded delation which leaves \(E\) pointwise fixed. \textit{S. L. Krushkal} [Sib. Math. J. 40, No. 4, 742--753 (1999; Zbl 0933.30014)] has shown that this can occur only if \(E\) has a quasicircle.NEWLINENEWLINENEWLINEIn the present paper the author solves the problem of finding the extremal quasiconformal reflection for the set \(E\) consisting of a circular arc and a point not on the circle bearing the arc. By normalization he reduces this to the case of the segment joining the points \(\pm 2i\) and a point on the positive real axis. His solution utilizes the \textit{O. Teichmüller} displacement theorem [Deutsche Math. 7, 336--343 (1944; Zbl 0060.23401)] which solves the following problem. For \(Q\)-quasiconformal mappings of the unit disc leaving all boundary points fixed find the region of values for the images of the centre. He uses this in a formulation due to \textit{E. Reich} [Ann. Acad. Sci. Fenn., Ser. A I, Math. 12, No. 2, 261--267 (1987; Zbl 0655.30016)]. He gives an explicit expression for the minimal/maximal dilation.