Schwarz's lemma

This article is about a complex calculus lemma. It should not be confused with Black's phrase .

The Schwarz Lemma (by Hermann Schwarz ) is a set of function theory about holomorphic self-images of the unit disk, which detects the zero point blank.

statement

Let it denote the open unit disk. Be a holomorphic function with . Then applies to all and . If in one point the equality also exists or applies, then a rotation, i.e. H. for a suitable . ${\displaystyle \mathbb {D} :=\left\{z\in \mathbb {C} \,:\,|z|<1\right\}}$ ${\displaystyle f\colon \mathbb {D} \to \mathbb {D} }$ ${\displaystyle f(0)=0}$ ${\displaystyle |f(z)|\leq |z|}$ ${\displaystyle z\in \mathbb {D} }$ ${\displaystyle |f'(0)|\leq 1}$ ${\displaystyle z_{0}\in \mathbb {D} ,z_{0}\neq 0,}$ ${\displaystyle |f(z_{0})|=|z_{0}|}$ ${\displaystyle |f'(0)|=1}$ ${\displaystyle f}$ ${\displaystyle f(z)=e^{i\lambda }\cdot z}$ ${\displaystyle \lambda \in \mathbb {R} }$

proof

Let be the Taylor expansion of around the point . Because is so the function ${\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}}$ ${\displaystyle f}$ ${\displaystyle z=0}$ ${\displaystyle f(0)=0}$ ${\displaystyle a_{0}=0}$

${\displaystyle g(z):={\begin{cases}{\frac {f(z)}{z}},&{\text{falls }}z\not =0,\\f'(0),&{\text{sonst}}\end{cases}}}$

is holomorphic and has the Taylor expansion around zero. After the maximum principle the function takes on the circle , their maximum on the edge of. The following applies there: ${\displaystyle \mathbb {D} }$ ${\displaystyle g(z)=\sum _{n=1}^{\infty }a_{n}z^{n-1}}$ ${\displaystyle g}$ ${\displaystyle K_{r}:=\{z\in \mathbb {C} \mid |z|\leq r\}}$ ${\displaystyle r\in (0,1)}$ ${\displaystyle \partial K_{r}=\{z\in \mathbb {C} \mid |z|=r\}}$

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

so that | g (z) | is limited to all through . Since is arbitrary, the border crossing already follows and thus for everyone . Furthermore is . ${\displaystyle K_{r}}$ ${\displaystyle {\frac {1}{r}}}$ ${\displaystyle 0<r<1}$ ${\displaystyle r\to 1}$ ${\displaystyle |g(z)|\leq 1}$ ${\displaystyle |f(z)|\leq |z|}$ ${\displaystyle z\in \mathbb {D} }$ ${\displaystyle |f'(0)|=|g(0)|\leq 1}$

Additionally, if one with exists or is true, then there is one with . With the maximum principle it follows that is constant, i.e. for a with . So it applies . ${\displaystyle z_{0}\in \mathbb {D} }$ ${\displaystyle |f(z_{0})|=|z_{0}|}$ ${\displaystyle |f'(0)|=1}$ ${\displaystyle z_{0}\in \mathbb {D} }$ ${\displaystyle |g(z_{0})|=1}$ ${\displaystyle g}$ ${\displaystyle g(z)=c}$ ${\displaystyle c}$ ${\displaystyle |c|=1}$ ${\displaystyle f(z)=c\cdot z}$

Applications

• Determination of holomorphic automorphism of the unit disc: . ${\displaystyle \mathrm {Aut} (\mathbb {D} )=\left\{f(z)=e^{i\lambda }{\frac {z-z_{0}}{1-{\overline {z_{0}}}z}}\;,\;\lambda \in [0,2\pi ),\;z_{0}\in \mathbb {D} \right\}}$

From this one can determine and obtain the automorphism group of the upper half-plane . ${\displaystyle \mathbb {H} }$ ${\displaystyle \mathrm {Aut} (\mathbb {H} )\cong PSL(2,\mathbb {R} )}$

• Schwarz's lemma is one of the tools used in the modern proof of Riemann's mapping theorem carried out with the help of normal families .
• Lemma von Schwarz-Pick : For holomorphic functions applies to all . ${\displaystyle f:\mathbb {D} \to \mathbb {D} }$ ${\displaystyle {\frac {|f'(z)|}{1-|f(z)|^{2}}}\leq {\frac {1}{1-|z|^{2}}}}$ ${\displaystyle z\in \mathbb {D} }$

Tightening

Among other things, Schwarz's lemma states that the condition applies to a holomorphic function with expansion in the power series . Ludwig Bieberbach showed that this also applies to injective functions , and put forward the Bieberbach hypothesis , later named after him , that for all . This conjecture was proven in 1985 by Louis de Branges de Bourcia . ${\displaystyle f:\mathbb {D} \to \mathbb {D} }$ ${\displaystyle f(0)=0}$ ${\displaystyle f(z)=\sum _{j=1}^{\infty }a_{j}z^{j}}$ ${\displaystyle |a_{1}|\leq 1}$ ${\displaystyle |a_{2}|\leq 2}$ ${\displaystyle |a_{j}|\leq j}$ ${\displaystyle j\in \mathbb {N} \;}$

literature

• Wolfgang Fischer, Ingo Lieb: Function theory. Vieweg Verlag, Braunschweig 2003, ISBN 3-528-77247-6