Asymptotic sharpness of a Bernstein-type inequality for rational functions in 𝐻²

Zarouf, R.

doi:10.1090/S1061-0022-2012-01198-2

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

St. Petersburg Math. J. 23 (2012), 309-319

DOI: https://doi.org/10.1090/S1061-0022-2012-01198-2

Published electronically: January 24, 2012

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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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