Schwarz lemma

The Schwarz Lemma (after Hermann Schwarz ) is a statement of the theory of functions holomorphic endomorphisms on the unit disk, which have a fixed point.

Statement

Denote the unit disk. Let be a holomorphic function with. Then already the case for all and. If there is equality or applicable in a point, it is already a rotation, ie for a fitting.

Evidence

Be the Taylor expansion of around the point. Because, such that the function of

Is holomorphic on and the Taylor expansion has around zero. According to the maximum principle, the function takes on the circle, its maximum on the boundary of. There, however, the following applies:

Such that | g ( z) | on all the way through is limited. Let us go to 1, it follows that and thus for all. Furthermore.

Applications

• Determination of holomorphic automorphisms of the unit disc.

• The Schwarz's lemma is one of the tools that are used in modern, conducted with the help of normal families proof of the Riemann mapping theorem.
• Lemma of Schwarz -Pick: applies to all for holomorphic functions.

Intensification

The Schwarz lemma implies among other things that the condition applies to a holomorphic function in the power series expansion. Ludwig Bieberbach showed that also applies to injective functions, and put on the later named after him Bieberbach conjecture that. This conjecture was proved in 1985 by Louis de Branges de Bourcia.