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ABSTRACT. This paper considers the rational system P n (a 1 Ò a 2 Ò    Ò a n ) :It gives a sharp (to constant) Markov-type inequality for real rational functions in P n (a 1 Ò a 2 Ò    Ò a n ). The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov-and Bernstein-type inequalities with curved majorants for rational functions in P n (a 1 Ò a 2 Ò    Ò a n ) are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results.

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