Schwarz's lemma

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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 .

proof

Let be the Taylor expansion of around the point . Because is so the function

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:

so that | g (z) | is limited to all through . Since is arbitrary, the border crossing already follows and thus for everyone . Furthermore is .

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 .

Applications

  • Determination of holomorphic automorphism of the unit disc: .
From this one can determine and obtain the automorphism group of the upper half-plane .
  • 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 .

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 .

literature

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