Another look at the Burns-Krantz theorem

Abstract

We obtain a generalization of the Burns-Krantz rigidity theorem for holomorphic self-mappings of the unit disk in the spirit of the classical Schwarz-Pick Lemma and its continuous version due to L.Harris via the generation theory for one-parameter semigroups. In particular, we establish geometric and analytic criteria for a holomorphic function on the disk with a boundary null point to be a generator of a semigroup of linear fractional transformations under some relations between three boundary derivatives of the function at this point.

Ahlfors-Beurling conformal invariant and relative capacity of compact sets

“…The results obtained here turn out to be effective in the study of a boundary version of the Schwarz lemma involving the Schwarzian derivative (see, e.g. [S,TV,D3]). After the completion of the writing of this paper, we learned about a very recent paper of S. Rohde and C. Wong, also studying half-plane capacity [RW].…”

“…For instance, Theorem 2.6 of the paper [TV] can now be understood as distortion result under the hypothesis that the image of the disk lies within a horocycle. In this context, there occurs also a natural connection with the initial coefficients of the expansion of the function in the neighborhood of a boundary point (see also [S,Theorem 4], [D3,Proposition 2]). Here a more general situation is considered, when the image of the disk has a property, characteristic for the application of the symmetrization method [H].…”

“…Some applications of Theorem 4.1 to the proof of geometric properties of holomorphic functions are given, for instance, in the following statements (cf. [S,TV,LLN,D3]).…”

Linear fractional mappings: invariant sets, semigroups and commutativity

“…In other words, this assertion does not cover the parabolic case. Nevertheless, one may state the following general assertion ( [35]). …”

Boundary behavior and rigidity of semigroups of holomorphic mappings

“…Comparing Theorems 1.2 and 1.3, we see again a cardinal difference between semigroups of hyperbolic and parabolic types: in the hyperbolic case with some smoothness conditions the limit curvature is always finite while in the parabolic case the limit curvature may be infinite. At the same time, it follows from [16] that if the second derivative f ′′ (1) is purely imaginary, then the third derivative f ′′′ (1) should be real. So, an immediate consequence of part (b) of Theorem 1.3 is the following fact.…”