IJPAM: Volume 30, No. 2 (2006)

QUASIDISKS AND THE MAX PROPERTY

Chu Yuming$^{1}$, Wang Gendi$^2$, Zhang Xiaohui$^3$
$^{1,2,3}$Department of Mathematics
Huzhou Teachers College
Huzhou, Zhejiang, 313000, P.R. CHINA
$^1$e-mail: chuyuming@hutc.zj.cn


Abstract.Let $D$ be a Jordan proper subdomain of $R^2$ whose boundary contains at least three points, $D^*=\overline R^2\setminus\overline D$, the exterior of $D$. We say that $D$ has the max property if there exists a constant $c\geq1$ such that each pair of points $x_1,x_2\in D\setminus\{\infty\}$ can be joined by an arc $\gamma$ in $D$ for which

\begin{displaymath}\vert x-y\vert\leq c\max_{j=1,2}\vert x_j-y\vert\end{displaymath}

for each $x\in\gamma$ and each $y\in\partial D$. In this paper, the authors prove that $D$ is a quasidisk if and only if both $D$ and $D^*$ have the max property.

Received: May 23, 2006

AMS Subject Classification: 30C62

Key Words and Phrases: quasiconformal mapping, quasidisk, max property

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2006
Volume: 30
Issue: 2