Abstract. 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-Szegő, V. S. Videnskiȋ, 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. §1. IntroductionThe Bernstein-Markov inequalitites for derivatives of trigonometric and algebraic polynomials, as well as their various generalizations, play an important role in approximation theory (see, e.g., the books [1, 2, 3] and also the recent papers [4,5,6,7] and the references therein).Our goal in the present paper is to obtain an inequality estimating the derivative of a real rational function that maps a given compact set E ⊂ R to another compact set F ⊂ R. We show that this inequality is sharp for any F and that it turns into the identity for a wide class of rational functions in the case where F is the union of two segments.We consider rational functions of degree n of the formwhere P n and Q n are real polynomials of degree at most n. Let x 1 , . . . , x n be the zeros of Q n (if deg Q n = n − k < n, we assume that x n−k+1 = · · · = x n = ∞). We shall use the harmonic measure ω(z, e, Ω) of a set e ⊂ E = ∂Ω at a point z relative to a domain Ω = C\E, and also the corresponding density). Next, we denote by Z n (x) the solution of the fourth Zolotarev problem (see [8,9,10]) concerning the best approximation of the function sgn x on E = [−1, κ]∪[κ, 1], 0 < κ < 1, by real rational functions of degree at most n.We cite one of the main results of [7].