A model space approach to some classical inequalities for rational functions

Abstract

We consider the set \mathcal{R}_{n} of rational functions of degree at most n\geq1 with no poles on the unit circle \mathbb{T} and its subclass \mathcal{R}_{n,\, r} consisting of rational functions without poles in the annulus \left\{\xi:\; r\leq|\xi|\leq\frac{1}{r}\right\}. We discuss an approach based on the model space theory which brings some integral representations for functions in \mathcal{R}_{n} and their derivatives. Using this approach we obtain L^{p}-analogs of several classical inequalities for rational functions including the inequalities by P. Borwein and T. Erd\'elyi, 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 \mathbb{T}.