On a Lebesgue constant of interpolation rational process at the Chebyshev – Markov nodes

In the present paper estimate of a Lebesgue constant of the interpolation rational Lagrange process on the segment $[-1,1]$, at the Chebyshev – Markov cosine fractions nodes is considered. It is shown that in the case of two real geometrically distinct poles of approximating functions, the norms of the Lagrange fundamental polynomials are bounded. Based on this result, it is proved that in the case under consideration the upper estimate of the Lebesgue constant does not depend on the arrangement of the poles and the sequence of the Lebesgue constant grows with logarithmic rate. Note, that in previous works the estimates of Lebesgue constants were obtained only for particular choices of poles or depended on the arrangement of poles.