IJPAM: Volume 34, No. 4 (2007)

REMARKS ON THE QUASIDISKS AND
THE HARDY-LITTLEWOOD PROPERTY

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


Abstract.Suppose that $D$ is a Jordan domain in the finit plane $R^2$, in this paper, the authors prove that $D$ is a quasidisk if and only if $f\in \text{\rm Lip}\,_{1,J}(D)$, whenever $f$ is analytic in $D$ with $\vert f'(z)\vert\leq d(z,\partial D)^{-1}$. Here $J(z_1,z_2)=\frac 12\log(1+\frac{\vert z_1-z_2\vert}{d(z_1,\partial D)})(1+\frac{\vert z_1-z_2\vert}{d(z_2,\partial
D)})$ for $z_1,z_2\in D$.

Received: November 13, 2006

AMS Subject Classification: 30C62

Key Words and Phrases: quasidisk, Hardy-Littlewood property, distance function

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 34
Issue: 4