ar X iv : m at h / 07 02 37 7 v 1 [ m at h . C V ] 1 3 Fe b 20 07 Another look at the Burns - Krantz Theorem
نویسنده
چکیده
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. Let ∆ be the open unit disk in the complex plane C and letHol(∆,Ω) be the set of all holomorphic functions (mappings) from ∆ into Ω ⊂ C. In particular, the set Hol (∆,∆) of all holomorphic self-mappings of ∆ is the semigroup with respect to composition operation. The famous rigidity theorem of D.M.Burns and S.G.Krantz ([7]) asserts: Let F ∈ Hol(∆,∆) be such that F (z) = 1 + (z − 1) +O (
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ar X iv : m at h / 06 02 37 1 v 1 [ m at h . A G ] 1 7 Fe b 20 06 Braid Monodromy of Hypersurface Singularities
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