Least deviation of logarithmic derivatives of algebraic polynomials from zero

We study least deviation of logarithmic derivatives of real-valued algebraic polynomials with a fixed root from zero on the segment [−1;1] in the uniform norm with various weights w(x). A final solution is given for w(x)=1−x2, and an asymptotically precise one is found for w(x)≡1. As a corollary, new inequalities of Markov-Bernstein type are obtained for a special class of polynomials.