A model space approach to some classical inequalities for rational functions

Abstract

We consider the set Rn of rational functions of degree at most n⩾1 with no poles on the unit circle T and its subclass Rn,r consisting of rational functions without poles in the annulus {ξ:r⩽|ξ|⩽1r}. We discuss an approach based on the model space theory which brings some integral representations for functions in Rn and their derivatives. Using this approach we obtain Lp-analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erdélyi, the Spijker Lemma and S.M. Nikolskii's inequalities. These inequalities are shown to be asymptotically sharp as n tends to infinity and the poles of the rational functions approach the unit circle T.