## Arkiv för Matematik

• Ark. Mat.
• Volume 37, Number 1 (1999), 171-182.

### Norm convergence of normalized iterates and the growth of Kœnigs maps

#### Abstract

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 Φnn converges uniformly on compact subsets of D to a function σ analytic in D which satisfies σ°φ=λσ. It is of interest in the study of composition operators to know if, whenever σ belongs to a Hardy space Hp, the sequence Φnn converges to σ in the norm of Hp. We show that this is indeed the case, generalizing a result of P. Bourdon obtained under the assumption that ϕ is univalent.

When ϕ is inner, P. Bourdon and J. Shapiro have shown that σ does not belong to the Nevanlinna class, in particular it does not belong to any Hp. It is natural to ask, how bad can the growth of σ be in this case? As a partial answer we show that σ always belongs to some Bergman space L ${}_{a}^{p}$ .

#### Note

The author is partially supported by NSF Grant DMS 97-06408 and wishes to thank Professor P. Bourdon for sharing results and conjectures.

#### Article information

Source
Ark. Mat., Volume 37, Number 1 (1999), 171-182.

Dates
First available in Project Euclid: 31 January 2017

https://projecteuclid.org/euclid.afm/1485898621

Digital Object Identifier
doi:10.1007/BF02384832

Mathematical Reviews number (MathSciNet)
MR1673430

Zentralblatt MATH identifier
1029.30017

Rights

#### Citation

Poggi-Corradini, Pietro. Norm convergence of normalized iterates and the growth of Kœnigs maps. Ark. Mat. 37 (1999), no. 1, 171--182. doi:10.1007/BF02384832. https://projecteuclid.org/euclid.afm/1485898621

#### References

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