Sharp quadrature formulas and inequalities between various metrics for rational functions

We obtain the sharp quadrature formulas for integrals of complex rational functions over circles, segments of the real axis, and the real axis itself. Among them there are formulas for calculating the $L_2$-norms of rational functions. Using the quadrature formulas for rational functions, in particular, for simple partial fractions and polynomials, we derive some sharp inequalities between various metrics (Nikol'skiĭ-type inequalities).