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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Mean growth of inner functions


Author: Domingo A. Herrero
Journal: Proc. Amer. Math. Soc. 37 (1973), 175-180
MSC: Primary 30A76
MathSciNet review: 0308403
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Abstract: Let A be a closed subset of the closed unit disc. It is shown that there exists a ``universal growth function'' $ \varphi (r,A)$ such that $ 1 - \smallint _0^{2\pi }\vert q(r{e^{ix}}){\vert^2}dx/2\pi = O(\varphi (r,A))$ for all inner functions $ q(z)$ whose zeroes lie in $ A \cap \{ \vert z\vert < 1\} $ and whose singularities in the unit circle lie on $ A \cap \{ \vert z\vert = 1\} $, if and only if the Lebesgue measure of $ A \cap \{ \vert z\vert = 1\} $ is zero.


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DOI: https://doi.org/10.1090/S0002-9939-1973-0308403-4
Keywords: Mean growth in $ {L^2}$-norm, inner function, Blaschke product, singular part, singular measure, singularities in the boundary
Article copyright: © Copyright 1973 American Mathematical Society