Estimate in a stability theorem of conformal maps of a disc (Q1080978)

The theory of stability of quasiconformal (qc) mappings deals with the problem of describing, in quantitative means, the nearness of a K-qc mapping to a conformal mapping when \(K\to 1\). The following theorem is proved. Theorem. For very K-qc mapping \(f: B\to B\) of the unit disc B onto itself there is a conformal automorphism \(\phi\) : \(B\to B\) such that \[ | f(z)-\phi (z)| \leq m \ln K \] for all \(z\in B\) where the least constant m satisfies the estimates \(1\leq m\leq 4/\pi =1.2732....\) The author has proved several related results recently [Sib. Mat. Zh. 17, 177-193 (1976; Zbl 0329.46035)] and [Dokl. Akad. Nauk SSSR 286, 295-297 (1986; reviewed below)].