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Another look at the BurnsKrantz theorem
 J. Anal. Math
, 2008
"... We obtain a generalization of the BurnsKrantz rigidity theorem for holomorphic selfmappings of the unit disk in the spirit of the classical SchwarzPick Lemma and its continuous version due to L.Harris via the generation theory for oneparameter semigroups. In particular, we establish geometric an ..."
Abstract

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We obtain a generalization of the BurnsKrantz rigidity theorem for holomorphic selfmappings of the unit disk in the spirit of the classical SchwarzPick Lemma and its continuous version due to L.Harris via the generation theory for oneparameter 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 let Hol(∆,Ω) be the set of all holomorphic functions (mappings) from ∆ into Ω ⊂ C. In particular, the set Hol (∆, ∆) of all holomorphic selfmappings 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
COMMON BOUNDARY REGULAR FIXED POINTS FOR HOLOMORPHIC SEMIGROUPS IN STRONGLY CONVEX DOMAINS
, 2014
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