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We consider the set \mathcal{R}_{n} of rational functions of degree at mostn\geq1 with no poles on the unit circle \mathbb{T} and its subclass\mathcal{R}_{n,\, r} consisting of rational functions without poles in theannulus \left\{\xi:\; r\leq|\xi|\leq\frac{1}{r}\right\}. We discuss an approachbased on the model space theory which brings some integral representations forfunctions in \mathcal{R}_{n} and their derivatives. Using this approach weobtain L^{p}-analogs of several classical inequalities for rational functionsincluding the inequalities by P. Borwein and T. Erd\'elyi, the Spijker Lemmaand S.M. Nikolskii's inequalities. These inequalities are shown to beasymptotically sharp as n tends to infinity and the poles of the rationalfunctions approach the unit circle \mathbb{T}.

A model space approach to some classical inequalities for rational functions