关键词: 拟共形映射, 交比, Poincare度量, 偏差

Abstract: In this paper, the author studies the distortion of cross ratio and poincar\'e metric under (1) If $f$ is a $k$-quasiconformal self
mapping of $\overline R^2$, then
$16^{\frac1k-1}\left(|(x_1,x_2,x_3,x_4)|+1\right)^{\frac1k}\leq | (f(x_1),f(x_2),f(x_3)$,
$f(x_4) ) |+1 \leq16^{k-1}\left(|(x_1,x_2,x_3,x_4)|+1\right)^{k}$ for any four points $x_1,x_2,x_3$,
$x_4\in\overline R^2$;

(2) If $f$ is a $k$-quasiconformal self mapping of $R^2$ and $D$ is a proper subdomain of $R^2$,
then $\frac1k\lambda_D(x_1,x_2)+4(\frac1k-1)\log2\leq\lambda_{f(D)}(f(x_1),f(x_2))\leq k\lambda_D(x_1,x_2)+4(k-1)\log2$
for any two points $x_1,x_2\in D$ plane quasiconformal mappings, obtaines the following two results.

Key words: Corss ratio, Poincare metric, Distortion； Quasiconformal mapping