Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 569-580

REMARKS ON ONE-COMPONENT INNER FUNCTIONS

Atte Reijonen

Tohoku University, Graduate School of Information Sciences
Aoba-ku, Sendai 980-8579, Japan; atte.reijonen 'at' uef.fi

Abstract. A one-component inner function Θ is an inner function whose level set

Ω_Θ(ε) = {z ∈ D : |Θ(z)| < ε}

is connected for some ε ∈ (0,1). We give a sufficient condition for a Blaschke product with zeros in a Stolz domain to be a one-component inner function. Moreover, a sufficient condition is obtained in the case of atomic singular inner functions. We study also derivatives of one-component inner functions in the Hardy and Bergman spaces. For instance, it is shown that, for 0 < p < ∞, the derivative of a one-component inner function Θ is a member of the Hardy space H^p if and only if Θ'' belongs to the Bergman space A_p-1^p, or equivalently Θ' ∈ A_p-1^2p.

2010 Mathematics Subject Classification: Primary 30J05; Secondary 30H10.

Key words: Bergman space, Blaschke product, Hardy space, one-component inner function, singular inner function.

Reference to this article: A. Reijonen: Remarks on one-component inner functions. Ann. Acad. Sci. Fenn. Math. 44 (2019), 569-580.

Full document as PDF file

https://doi.org/10.5186/aasfm.2019.4434