Schwarz-Pick's Lemma

The lemma of Schwarz-Pick (by Hermann Schwarz and Georg Alexander Pick ) is a statement from the function theory about holomorphic endomorphisms of the unit circle, which the Schwarz Lemma generalized. In the context of hyperbolic geometry , it means that holomorphic endomorphisms are contractions .

statement

Let it denote the unit disk and be a holomorphic function. Then applies to everyone ${\displaystyle {\mathbb {D} }:=\left\{z\in \mathbb {C} \,:\,|z|<1\right\}}$ ${\displaystyle f\colon \mathbb {D} \to \mathbb {D} }$ ${\displaystyle z_{1},z_{2}\in {\mathbb {D} },z_{1}\not =z_{2}}$

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

and for everyone ${\displaystyle z\in {\mathbb {D} }}$

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

The second statement follows from the first by dividing by and then letting go against . ${\displaystyle |z_{1}-z_{2}|}$ ${\displaystyle z_{1}}$ ${\displaystyle z_{2}}$

Applications

In hyperbolic geometry is

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

the hyperbolic distance. The first inequality of Schwarz-Pick's lemma says that holomorphic functions with regard to this metric are contractions . ${\displaystyle {\mathbb {D} }\rightarrow {\mathbb {D} }}$

If and if one sets in the first inequality , then one receives the statement of Schwarz's lemma as a special case . ${\displaystyle f(0)=0}$ ${\displaystyle z_{2}=0}$

literature

• Wolfgang Fischer, Ingo Lieb : Function theory . Vieweg, Braunschweig et al. 1980, ISBN 3-528-07247-4 , ( Vieweg-Studium 47: Advanced course in mathematics ).

Web links

• Georg Pick On a property of the conformal mapping of circular areas , Mathematische Annalen, Vol. 77, 1916, pp. 1–6
• Osserman "From Schwarz-Pick to Ahlfors and beyond", Notices American Mathematical Society, August 1999, as a PDF file here [1] (PDF; 90 kB)