Extremal functions for plane quasiconformal mappings

Abstract

For the family $\mathscr{F}(K)$ of $K$-quasiconformal mappings $f$ from $\mathbb{\bar{C}} = \{|z|\leqslant +\infty \}$ onto $\mathbb{C}$ such that $f(\mathbb{R}) = \mathbb{R}$ and $f(x) = x$ for $x=-1$, $0$, $\infty$, the supremum $\lambda (K, t)$ and the infimum $\nu (K, t)$ of $f(t)$ for $f$ ranging over $\mathscr{F}(K)$ with $t \in \mathbb{R}$ fixed are studied. They are expressed by the inverse $\mu ^{-1}$ of the function $\mu (r)$, the modulus of the bounded, doubly-connected domain with the unit circle and the real interval $[0, r]$, $0 < r < 1$, as the boundary. Among a number of results obtained, asymptotic behaviors of $X(K, t)(X = \lambda , \nu )$ as $t \to \pm \infty$ for a fixed $K$ and as $K \to +\infty$ for a fixed $t$ are considered.