Estimates for derivatives of rational functions and the fourth Zolotarev problem

An estimate is obtained for the derivatives of real rational functions that map a compact set on the real line to another set of the same kind. Many well-known inequalities (due to Bernstein, Bernstein—Szegö, V. S. Videnskii, V. N. Rusak, and M. Baran–V. Totik) are particular cases of this estimate. It is shown that the estimate is sharp. With the help of the solution of the fourth Zolotarev problem, a class of examples is constructed in which the estimates obtained turn into identities.