INEQUALITIES FOR RATIONAL FUNCTIONS WITH PRESCRIBED POLES

This paper considers the rational system P-n(a(1), a(2),..., a(n)); = {P(x)/Pi(k=1)(n) (x-a(k)), P is an element of P-n} with nonreal elemen ts in {a(k)}(k=1)(n) subset of C / [-1, 1] paired by complex conjugati on. It gives a sharp (to constant) Markov-type inequality for real rat ional functions in P-n(a(1), a(2),..., a(n)). The corresponding Markov -type inequality for high derivatives is established, as well as Nikol skii-type inequalities. Some sharp Markov-and Bernstein-type inequalit ies with curved majorants for rational functions in P-n(a(1), a(2),... , 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.