# 施瓦茨引理

${\displaystyle \mathbb {D} =\{z:|z|<1\}}$複平面中的开圆盘，如果

• ${\displaystyle f:\mathbb {D} \to \mathbb {C} }$是全纯函数、
• ${\displaystyle f(0)=0}$
• ${\displaystyle |f(z)|\leq 1}$

${\displaystyle |f(z)|\leq |z|}$

${\displaystyle |f(z)|=|z|\,}$

${\displaystyle |f'(0)|=1\,}$

## 证明

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

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

## 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${\displaystyle z_{1},z_{2}\in \mathbb {H} }$

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

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

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

## 参考

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