## Journal of Mathematics of Kyoto University

### 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.

#### Article information

Source
J. Math. Kyoto Univ., Volume 43, Number 1 (2003), 71-99.

Dates
First available in Project Euclid: 14 August 2009

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1250283741

Digital Object Identifier
doi:10.1215/kjm/1250283741

Mathematical Reviews number (MathSciNet)
MR2028701

Zentralblatt MATH identifier
1064.30010

#### Citation

Kurihara, Shigenori; Yamashita, Shinji. Extremal functions for plane quasiconformal mappings. J. Math. Kyoto Univ. 43 (2003), no. 1, 71--99. doi:10.1215/kjm/1250283741. https://projecteuclid.org/euclid.kjm/1250283741