It is considered the Lagrange interpolation processes such that rational functions with fixed denominators play the role of polynomials vanishing at interpolation nodes. An estimate for Lebesgue constants is obtained for the case of rational functions deviated least from zero on a given system of intervals with maximally possible number of deviation points, and when the matrix of fixed poles is contained in a compact set outside of the system of intervals. V. N. Rusak and G. Min found earlier particular case (for the case of one interval).
