Inequalities for Rational Functions With Prescribed Poles

Min, G.

Canadian journal of mathematics, Tome 50 (1998) no. 1, pp. 152-166

Résumé

This paper considers the rational system ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})\,\,:=\,\left\{ \frac{P(x)}{\Pi _{k=1}^{n}(x-{{a}_{k}})},\,P\,\in \,{{P}_{n}} \right\}$ with nonreal elements in $\left\{ {{a}_{k}} \right\}_{k=1}^{n}\,\subset \,\mathbb{C}\,\backslash \,[-1,\,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$ . 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 ${{P}_{n}}({{a}_{1}},{{a}_{2}},...,{{a}_{n}})$ 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

Détail

DOI : 10.4153/CJM-1998-008-3

Mots-clés : 41A17, 26D07, 26C15, Markov-type inequality, Bernstein-type inequality, Nikolskii-type inequality, Schur-type inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials

Bibliographie

[1] 1. Achiezer, N.I., Theory of Approximation.Ungar, New York, 1956. Google Scholar

[2] 2. Borwein, P., Inequalities and inverse theorems in restricted rational approximation theory. Canad. J. Math. 32 (1980), 345–361. Google Scholar

[3] 3. Borwein, P. and Erdélyi, T., Polynomials and Polynomial Inequalities.Springer, New York, 1995. Google Scholar

[4] 4. Borwein, P. , Sharp extensions of Bernstein's inequality to rational spaces. Mathematica 43 (1996), 413–423. Google Scholar

[5] 5. Borwein, P., Erdélyi, T. and Zhang, J., Chebyshev polynomials and Markov-Bernstein type inequalities for the rational spaces. J. London Math. Soc. 50 (1994), 501–519. Google Scholar

[6] 6. Cheney, E.W., Introduction to Approximation Theory.McGraw-Hill, New York, 1966. Google Scholar

[7] 7. Lachance, M.A., Bernstein and Markov inequalities for constrained polynomials. Rational Approximation and Interpolation, 125–135. Lecture Notes in Math. , Springer-Verlag, 1984. Google Scholar

[8] 8. Li, X., Mohapatra, R.N. and Rodriguez, R.S., Bernstein-type inequalities for rational functions with prescribed poles. J. London Math. Soc. 51 (1995), 523–531. Google Scholar

[9] 9. Lorentz, G.G., Approximation of Functions.Holt Rinehart and Winston, New York, 1966. Google Scholar

[10] 10. Milovanovi, G.V.ć, Mitrinovi, D.S.ć and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore, 1994. Google Scholar

[11] 11. Natanson, I.P., Constructive Function Theory. Vol I, Ungar, New York, 1964. Google Scholar

[12] 12. Petrushev, P. and Popov, V.A., Rational Approximation of Real Functions. Cambridge Univ. Press, 1987. Google Scholar

[13] 13. Rahman, Q.I., On a problem of Turán about polynomials with curved majorants. Trans. Amer. Math. Soc. 163 (1972), 447–455. Google Scholar

[14] 14. Rahman, Q.I. and Schmeisser, G., Les Inégalités de Markoff et de Bernstein. Les Presses de L’Université de Montreal, 1983. Google Scholar

[15] 15. Rivlin, T.J., Chebyshev Polynomials.second edition, John Wiley & Sons, 1990. Google Scholar