施瓦茨引理

陈述

${\displaystyle \mathbb {D} =\{z:|z|<1\}}$ 是复平面上以原点为圆心的单位开圆盘。全纯函数${\displaystyle f:\mathbb {D} \to \mathbb {D} }$ 满足${\displaystyle f(0)=0}$ ，则对任意${\displaystyle z\in \mathbb {D} }$ ${\displaystyle |f(z)|<1}$ ${\displaystyle |f'(0)|\leq 1}$ 。此外，如果存在${\displaystyle z}$ 使得${\displaystyle |f(z)|=|z|}$ ，或者${\displaystyle |f'(0)|=1}$ ，则f是一个旋转${\displaystyle f(z)=az}$ ，其中${\displaystyle |a|=1}$

证明

${\displaystyle g(z)={\begin{cases}{\frac {f(z)}{z}}\,&{\mbox{if }}z\neq 0\\f'(0)&{\mbox{if }}z=0,\end{cases}}}$

${\displaystyle |g(z)|\leq |g(z_{r})|={\frac {|f(z_{r})|}{|z_{r}|}}\leq {\frac {1}{r}}}$

r趋于1时，得到|g(z)| ≤ 1。

施瓦茨-皮克定理

${\displaystyle f:\mathbb {D} \to \mathbb {D} }$  全纯。那么，对所有${\displaystyle z_{1},z_{2}\in \mathbb {D} }$

${\displaystyle \left|{\frac {f(z_{1})-f(z_{2})}{1-{\overline {f(z_{1})}}f(z_{2})}}\right|\leq {\frac {\left|z_{1}-z_{2}\right|}{\left|1-{\overline {z_{1}}}z_{2}\right|}}}$

${\displaystyle {\frac {\left|f'(z)\right|}{1-\left|f(z)\right|^{2}}}\leq {\frac {1}{1-\left|z\right|^{2}}}.}$

${\displaystyle d(z_{1},z_{2})=\tanh ^{-1}\left({\frac {\left|z_{1}-z_{2}\right|}{\left|1-{\overline {z_{1}}}z_{2}\right|}}\right)}$

${\displaystyle f:\mathbb {H} \to \mathbb {H} }$ 全纯。那么，对所有${\displaystyle z_{1},z_{2}\in \mathbb {H} }$

${\displaystyle \left|{\frac {f(z_{1})-f(z_{2})}{{\overline {f(z_{1})}}-f(z_{2})}}\right|\leq {\frac {\left|z_{1}-z_{2}\right|}{\left|{\overline {z_{1}}}-z_{2}\right|}}}$

${\displaystyle {\frac {\left|f'(z)\right|}{{\mbox{Im }}f(z)}}\leq {\frac {1}{{\mbox{Im }}z}}.}$

${\displaystyle f(z)={\frac {az+b}{cz+d}}}$

施瓦茨-皮克定理的证明

${\displaystyle {\frac {z-z_{0}}{{\bar {z}}_{0}z-1}},|z_{0}|<1}$

${\displaystyle M(z)={\frac {z_{1}-z}{1-{\bar {z}}_{1}z}},\varphi (z)={\frac {f(z_{1})-z}{1-{\overline {f(z_{1})}}z}}}$

${\displaystyle |\varphi \circ f\circ M^{-1}(z)|=\left|{\frac {f(z_{1})-f(M^{-1}(z))}{1-{\overline {f(z_{1})}}f(M^{-1}(z))}}\right|\leq |z|}$

${\displaystyle z_{2}=M^{-1}(z)}$ ，就得到想要的结论

${\displaystyle \left|{\frac {f(z_{1})-f(z_{2})}{1-{\overline {f(z_{1})}}f(z_{2})}}\right|\leq {\frac {\left|z_{1}-z_{2}\right|}{\left|1-{\overline {z_{1}}}z_{2}\right|}}}$

${\displaystyle \left|{\frac {f(z_{1})-f(z_{2})}{z_{1}-z_{2}}}\right|\leq \left|{\frac {1-{\overline {f(z_{1})}}f(z_{2})}{1-{\bar {z}}_{1}z_{2}}}\right|}$

${\displaystyle z_{2}}$ 趋向于${\displaystyle z_{1}}$ 即得。

进一步的推广与相关结果

De Brange定理，以前称为Bieberbach猜想，是该引理的一个重要推广。

Koebe四分之一定理，给出了f是单值的情况下的一个相关的估计。

参考

• Jurgen Jost, Compact Riemann Surfaces (2002), Springer-Verlag, New York. ISBN 3-540-43299-X (See Section 2.3)