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Proceedings of the American Mathematical Society

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On a theorem of Privalov and normal functions

Author: Daniel Girela
Journal: Proc. Amer. Math. Soc. 125 (1997), 433-442
MSC (1991): Primary 30D45, 30D55
MathSciNet review: 1363422
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Abstract: A well known result of Privalov asserts that if $f$ is a function which is analytic in the unit disc $\Delta =\{z\in \mathbb {C} : \vert z\vert <1\} $, then $f$ has a continuous extension to the closed unit disc and its boundary function $f(e\sp {i\theta })$ is absolutely continuous if and only if $f\sp {\prime }$ belongs to the Hardy space $H\sp {1}$. In this paper we prove that this result is sharp in a very strong sense. Indeed, if, as usual, $M_{1}(r, f\sp {\prime })= \frac {1}{2\pi }\int _{-\pi }\sp {\pi }\left \vert f\sp {\prime }(re\sp {i\theta }) \right \vert \, d\theta ,$ we prove that for any positive continuous function $\phi $ defined in $(0, 1)$ with $\phi (r)\to \infty $, as $r\to 1$, there exists a function $f$ analytic in $\Delta $ which is not a normal function and with the property that $M_{1}(r, f\sp {\prime })\leq \phi (r) $, for all $r$ sufficiently close to $1$.

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Daniel Girela
Affiliation: Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, 29071 Málaga, Spain

Keywords: Normal functions, Hardy spaces, integral means, theorem of Privalov
Received by editor(s): November 1, 1994
Received by editor(s) in revised form: June 25, 1995
Additional Notes: This research has been supported in part by a D.G.I.C.Y.T. grant (PB91-0413) and by a grant from “La Junta de Andalucía”
Communicated by: Albert Baernstein II
Article copyright: © Copyright 1997 American Mathematical Society