  We study least deviation of logarithmic derivatives of real-valued
polynomials with a fixed root from zero on the segment $[-1;1]$ in the uniform
norm with the weight $\sqrt{1-x^2}$ and without it. Basing on results of
Komarov and Novak and on a certain determinant identity due to Borchardt, we
also establish a criterion for best uniform approximation of continuous
real-valued functions by logarithmic derivatives in terms of a Chebyshev
alternance.
