St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

   
 

 

Asymptotic sharpness of a Bernstein-type inequality for rational functions in $ H^2$


Author: R. Zarouf
Original publication: Algebra i Analiz, tom 23 (2011), nomer 2.
Journal: St. Petersburg Math. J. 23 (2012), 309-319
MSC (2010): Primary 30H10, 30J10
Published electronically: January 24, 2012
MathSciNet review: 2841678
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Abstract: A Bernstein-type inequality for the standard Hardy space $ H^2$ in the unit disk $ \mathbb{D}=\{z\in \mathbb{C}\,:\,\vert z\vert <1\}$ is considered for rational functions in $ \mathbb{D}$ having at most $ n$ poles all outside of $ \frac {1}{r}\mathbb{D}$, $ 0<r<1$. The asymptotic sharpness is shown as $ n\rightarrow \infty $ and $ r\rightarrow 1$.


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Additional Information

R. Zarouf
Affiliation: CMI-LATP, UMR 6632, Université de Provence, 39 rue F.-Joliot-Curie, 13453 Marseille cedex 13, France
Email: rzarouf@cmi.univ-mrs.fr

DOI: http://dx.doi.org/10.1090/S1061-0022-2012-01198-2
Keywords: Bernstein inequality, finite Blaschke product, Hardy space
Received by editor(s): January 29, 2010
Published electronically: January 24, 2012
Article copyright: © Copyright 2012 American Mathematical Society