# 博雷尔-卡拉西奥多里定理

## 定理陈述

${\displaystyle \|f\|_{r}\leq {\frac {2r}{R-r}}\sup _{|z|\leq R}{\operatorname {Re} {f(z)}}+{\frac {R+r}{R-r}}|f(0)|}$

${\displaystyle \|f\|_{r}=\max _{|z|\leq r}{|f(z)|}=\max _{|z|=r}{|f(z)|}}$

## 证明

${\displaystyle w\mapsto w/A-1}$ ${\displaystyle P}$ 变成标准左半平面。${\displaystyle w\mapsto R(w+1)/(w-1)}$ 把左半平面变成圆心在原点且半径为${\displaystyle R}$ 的圆。它们的复合映射把0映成0，就是所需要的映射：

${\displaystyle w\mapsto {\frac {Rw}{w-2A}}}$

${\displaystyle {\frac {|Rf(z)|}{|f(z)-2A|}}\leq |z|}$

${\displaystyle |z| ，上式变为

${\displaystyle R|f(z)|\leq r|f(z)-2A|\leq r|f(z)|+2Ar}$

${\displaystyle |f(z)|\leq {\frac {2Ar}{R-r}}}$

{\displaystyle {\begin{aligned}|f(z)|-|f(0)|&\leq |f(z)-f(0)|\\&\leq {\frac {2r}{R-r}}\sup _{|w|\leq R}{\operatorname {Re} {(f(w)-f(0))}}\\&\leq {\frac {2r}{R-r}}\left(\sup _{|w|\leq R}{\operatorname {Re} {f(w)}+|f(0)|}\right)\\\end{aligned}}}

## 参考资料

• Lang, Serge (1999). Complex Analysis (4th ed.). New York: Springer-Verlag, Inc. ISBN 0-387-98592-1.
• Titchmarsh, E. C. (1938). The theory of functions. Oxford University Press.