%0 Journal Article
%T A Bernstein-type inequality for rational functions in weighted Bergman spaces
%+ Department of Mathematics and Mechanics
%+ Mathématiques Fondamentales
%A Baranov, Anton
%A Zarouf, Rachid
%< avec comité de lecture
%@ 0007-4497
%J Bulletin des Sciences Mathématiques
%I Elsevier
%V 137
%P 541--556
%8 2013-08-01
%D 2013
%Z 1003.5066
%K Rational function
%K Bernstein-type inequality
%K weighted Bergman norm
%Z Primary 32A36, 26A33; Secondary 26C15, 41A10
%Z Mathematics [math]/Functional Analysis [math.FA]
%Z Mathematics [math]/Complex Variables [math.CV]Journal articles
%X Given $n\geq1$ and $r\in[0,\,1),$ we consider the set $\mathcal{R}_{n,\, r}$ of rational functions having at most $n$ poles all outside of $\frac{1}{r}\mathbb{D},$ were $\mathbb{D}$ is the unit disc of the complex plane. We give an asymptotically sharp Bernstein-type inequality for functions in $\mathcal{R}_{n,\, r}\:$ (as n tends to infinity and r tends to 1-) in weighted Bergman spaces with ''polynomially'' decreasing weights. We also prove that this result can not be extended to weighted Bergman spaces with ''super-polynomially'' decreasing weights.
%G English
%2 https://hal.science/hal-00463664v4/document
%2 https://hal.science/hal-00463664v4/file/bergm-bernst-27-June.pdf
%L hal-00463664
%U https://hal.science/hal-00463664
%~ LATP
%~ CNRS
%~ UNIV-AMU
%~ EC-MARSEILLE
%~ I2M
%~ I2M-2014-
%~ ESPE-AMU_PUBLICATIONS