Extremal functions for plane quasiconformal mappings (Q1890270)

Let \( F(K) \) be the family of \(K\)-quasiconformal mappings \(f\) from the extended complex plane \( \overline{C} = C \cup \{\infty \}\) onto \(\overline{C} \) such that \(f(R) = R \) and \(f(-1)=-1, f(0) = 0, f(\infty)= \infty.\) The aim of the paper are the following extremal problems for a fixed \(t \in R \) \[ \lambda(K,t) = \sup_{f \in F(K)}f(t), \quad \nu(K,t) = \inf_{f \in F(K) } f(t). \] Basing on a method introduced by \textit{O. Lehto, K. I. Virtanen} and \textit{J. Väisälä} [Ann. Acad. Sci. Fenn., Ser. A I 273, 14 p. (1959; Zbl 0090.05102)] the authors determine the expressions for \(\lambda\) and \(\nu\), which are given in terms of the modulus of the Grötzsch ring. The paper contains also asymptotic results for \(t \to \infty \) and among a number of other results a chapter where the hyperbolic distance has been considered. For the last mentioned topic a reference to \textit{A. Yu. Solynin} and \textit{M. Vuorinen} [Isr. J. Math. 124, 29--60 (2001; Zbl 1021.30047)] would be appropriate.