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On Blaschke products, Bloch functions and normal functions

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Abstract

We prove that if G is an analytic function in the unit disc such that G(z)→∞, as z→1, and B is an infinite Blaschke product whose sequence of zeros is contained in a Stolz angle with vertex at 1 then the function f=B⋅G is not a normal function.

We prove also some results on the asymptotic cluster set of a thin Blaschke product with positive zeros which are related with the question of the existence of non-normal outer functions with restricted mean growth of the derivative.

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Authors and Affiliations

  1. Departamento de Análisis Matemático, Universidad de Málaga, Campus de Teatinos, 29071, Málaga, Spain

    Daniel Girela

  2. Departament de Matemátiques, Universitat Autónoma de Barcelona, Bellaterra, 08193, Barcelona, Spain

    Daniel Suárez

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  1. Daniel Girela
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  2. Daniel Suárez
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Correspondence to Daniel Girela.

Additional information

The authors thankfully acknowledge partial support from the following grants. The first author: MTM2007-60854 (MICINN, Spain ); FQM-210 and P06-FQM01504 (Junta de Andalucía) and “Harmonic and Complex Analysis and Its Applications” (European Networking Programme, ESF). The second author: The Ramón and Cajal program and grants MTM2008-00145 and 2009SGR00420 (MICINN, Spain ). Both authors: MTM2008-02829-E and Ingenio Mathematica (i-MATH) No. CSD2006-00032 (MICINN, Spain).

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Girela, D., Suárez, D. On Blaschke products, Bloch functions and normal functions. Rev Mat Complut 24, 49–57 (2011). https://doi.org/10.1007/s13163-010-0027-6

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