Inequalities for derivatives of rational functions on several intervals

Estimates for derivatives of rational functions and the fourth Zolotarev problem

“…This theorem was presented in [7] without proof (and was formulated without the equality case), but its justification practically repeats the proof of [7,Theorem 4]. The only modification is that, for replacement of a "zero" at infinity, the polynomial in question should be multiplied by a factor of the form…”

“…In its turn, this yields a series of known inequalities (a more detailed account of the history of this topic can be found in [7]). …”

“…§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 , . .…”

“…L. LUKASHOV Remark 2. By using [7, Theorems 2 and 4], it is possible to prove a result similar to Theorem 2 for the ratio of trigonometric polynomials. This result was announced in [23].…”

“…The 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).…”