Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 43, 2018, 1023-1026

OUTER FUNCTIONS AND UNIFORM INTEGRABILITY

Javad Mashreghi and Thomas Ransford

Université Laval, Département de mathématiques et de statistique
Québec City (Québec), Canada G1V 0A6; javad.mashreghi 'at' mat.ulaval.ca

Université Laval, Département de mathématiques et de statistique
Québec City (Québec), Canada G1V 0A6; thomas.ransford 'at' mat.ulaval.ca

Abstract. We show that, if f is an outer function and a ∈ [0,1), then the set of functions

{log|(f ˆ ψ)^*| : ψ : D → D holomorphic, |ψ(0)| ≤ a}

is uniformly integrable on the unit circle. As an application, we derive a simple proof of the fact that, if f is outer and φ : D → D is holomorphic, then f : φ is outer.

2010 Mathematics Subject Classification: Primary 30H15; Secondary 28A20.

Key words: Outer function, Smirnov class, uniformly integrable.

Reference to this article: J. Mashreghi and T. Ransford: Outer functions and uniform integrability. Ann. Acad. Sci. Fenn. Math. 43 (2018), 1023-1026.

Full document as PDF file

https://doi.org/10.5186/aasfm.2018.4360