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We obtain new parametric quadrature formulas with variable nodes forintegrals of complex rational functions over circles, segments of the real axisand the real axis itself. Basing on these formulas we derive$(q,p)$-inequalities of Jackson-Nikolskii type for various classes of rationalfunctions, complex polynomials and their logarithmic derivatives (simplepartial fractions). It is shown that our $(\infty,2)$- and$(\infty,4)$-inequalities are sharp in a number of main theorems. Ourinequalities extend and refine several results obtained earlier by otherauthors.

Quadrature formulas with variable nodes and Jackson-Nikolskii inequalities for rational functions