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