On a problem of Turan about polynomials
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- by R. Pierre and Q. I. Rahman
- Proc. Amer. Math. Soc. 56 (1976), 231-238
- DOI: https://doi.org/10.1090/S0002-9939-1976-0412362-6
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Abstract:
It is shown that if ${p_n}(x)$ is a polynomial of degree $n$ whose graph on the interval $- 1 < x < 1$ is contained in the unit disk then the absolute value of its second derivative cannot exceed $\tfrac {2}{3}(n - 1)(2{n^2} - 4n + 3)$ on $[ - 1,1]$.References
- Morris Marden, Geometry of polynomials, 2nd ed., Mathematical Surveys, No. 3, American Mathematical Society, Providence, R.I., 1966. MR 0225972 A. A. Markoff, On a problem of D. I. Mendeleev, Zap. Imp. Akad. Nauk 62 (1889), 1-24. (Russian) W. A. Markoff, Über Polynome, die in einem gegebenen Intervalle möglichst wenig von Null abweichen, Math. Ann. 77 (1916), 218-258.
- Q. I. Rahman, Functions of exponential type, Trans. Amer. Math. Soc. 135 (1969), 295–309. MR 232938, DOI 10.1090/S0002-9947-1969-0232938-X
- Q. I. Rahman, On a problem of Turán about polynomials with curved majorants, Trans. Amer. Math. Soc. 163 (1972), 447–455. MR 294586, DOI 10.1090/S0002-9947-1972-0294586-5
Bibliographic Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 56 (1976), 231-238
- MSC: Primary 26A75
- DOI: https://doi.org/10.1090/S0002-9939-1976-0412362-6
- MathSciNet review: 0412362