Journal of Mathematics of Kyoto University

Extremal functions for plane quasiconformal mappings

Shigenori Kurihara and Shinji Yamashita

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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

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

First available in Project Euclid: 14 August 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30C62: Quasiconformal mappings in the plane
Secondary: 30C75: Extremal problems for conformal and quasiconformal mappings, other methods


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.

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