Norm convergence of normalized iterates and the growth of Kœnigs maps

Pietro Poggi-Corradini Department of Mathematics Cardwell Hall, Kansas State University

TBD mathscidoc:1701.332910

Arkiv for Matematik, 37, (1), 171-182, 1997.8
Let ϕ be an analytic function defined on the unit disk$D$, with ϕ($D$)⊂$D$, ϕ(0)=0, and ϕ′(0)=λ≠0. Then by a classical result of G. Kœnigs, the sequence of normalized iterates Φ_{$n$}/λ^{$n$}converges uniformly on compact subsets of$D$to a function σ analytic in$D$which satisfies$σ$°φ=λ$σ$. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space$H$_{$p$}, the sequence Φ_{$n$}/λ^{$n$}converges to σ in the norm of$H$_{$p$}. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.
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@inproceedings{pietro1997norm,
  title={Norm convergence of normalized iterates and the growth of Kœnigs maps},
  author={Pietro Poggi-Corradini},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203555344824719},
  booktitle={Arkiv for Matematik},
  volume={37},
  number={1},
  pages={171-182},
  year={1997},
}
Pietro Poggi-Corradini. Norm convergence of normalized iterates and the growth of Kœnigs maps. 1997. Vol. 37. In Arkiv for Matematik. pp.171-182. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203555344824719.
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