Geodesic
On a Problem of Turán about Polynomials II
Canadian journal of mathematics, Tome 33 (1981) no. 3, pp. 701-733

Voir la notice de l'article provenant de la source Cambridge University Press

1. It was proved by A. A. Markov [3] that if is a polynomial of degree at most n and |pn (x)| ≦ 1 in the interval –1 ≦ x ≦ 1, then in the same interval (1) The problem was proposed by the chemist Mendeleev who knew the answer for polynomials of degree 2. For a historical background of the problem see [1].A. A. Markov's younger brother W. A. Markov considered the problem of determining exact bounds for the j–th derivative of pn (x) at a given point X 0 in [ – 1, 1]. His results appeared in a Russian journal in the year 1892; a German version of his remarkable paper was later published in [4]. Amongst other things he proved the following two theorems. 
Pierre, R.; Rahman, Q. I. On a Problem of Turán about Polynomials II. Canadian journal of mathematics, Tome 33 (1981) no. 3, pp. 701-733. doi: 10.4153/CJM-1981-055-8
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[1] 1. Boas, R. P., Jr., Inequalities for the derivatives of polynomials, Math. Mag. 42 (1969), 165–174. Google Scholar

[2] 2. Dzyadyk, V. K., On a constructive characteristic of functions satisfying the Lipschitz condition a (0 < a <1), on a finite segment of the real axis, Izv. Akad. Nauk SSSR (seriya mat.) 20 (1956), 623–642. Google Scholar

[3] 3. Markov, A. A., On a problem of D. I. Mendeleev, Zap. Imp. Akad. Nauk 62 (1889), 1–24. Google Scholar

[4] 4. Markov, W. A., Uber Polynôme, die in einem gegebenen Intervalle moglichst wenig von Null abweichen, Math. Ann. 77 (1916), 218–258. Google Scholar

[5] 5. Pierre, R. and Rahman, Q. I., On a problem of Turan about polynomials, Proc. Amer. Math. Soc. 56 (1976), 231–238. Google Scholar

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

[7] 7. Rahman, Q. I., Addendum to uOn a problem of Turdn about polynomials with curved majorants”, Trans. Amer. Math. Soc. 168 (1972), 517–518. Google Scholar

[8] 8. Szegô, G., Orthogonal polynomials, American Mathematical Society Colloquium Publications 23 (American Mathematical Society, Providence, Rhode Island, Third edition, 1967). Google Scholar

[9] 9. Voronovskaja, E. V., The functional method and its applications, Translations of Mathematical Monographs 28 (American Mathematical Society, Providence, Rhode Island, 1970). Google Scholar

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