Inequalities for the Derivatives of Rational Functions with Real Zeros

We study Turán-type inequalities for the derivatives of rational functions, whose zeros are all real and lie inside -1,1] but whose poles are outside (-1,1), in the supremum- and L2-norms respectively. We generalize several well-known results for classical polynomials. We also obtain a sharp L2 Bernstein-type inequality for the rational system with prescribed poles.