Markov’s inequality for polynomials with real zeros
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- by Peter Borwein PDF
- Proc. Amer. Math. Soc. 93 (1985), 43-47 Request permission
Abstract:
Markov’s inequality asserts that $||{p’_n}|| \leqslant {n^2}||{p_n}||$ for any polynomial ${p_n}$ of degree $n$. (We denote the supremum norm on $[ - 1,1]$ by $||.||$.) In the case that ${p_n}$ has all real roots, none of which lie in $[ - 1,1]$, Erdös has shown that $||{p’_n}|| \leqslant en||{p_n}||/2$. We show that if ${p_n}$ has $n - k$ real roots, none of which lie in $[ - 1,1]$, then $||{p’_n} \leqslant cn(k + 1)||{p_n}||$, where $c$ is independent of $n$ and $k$. This extension of Markov’s and Erdös’ inequalities was conjectured by Szabados.References
- P. Erdös, On extremal properties of the derivatives of polynomials, Ann. of Math. (2) 41 (1940), 310–313. MR 1945, DOI 10.2307/1969005
- G. G. Lorentz, Approximation of functions, Holt, Rinehart and Winston, New York-Chicago, Ill.-Toronto, Ont., 1966. MR 0213785 A. A. Markov, On a problem of D. I. Mendeleev, Acad. Sci., St. Petersburg 62 (1889), 1-24.
- Attila Máté, Inequalities for derivatives of polynomials with restricted zeros, Proc. Amer. Math. Soc. 82 (1981), no. 2, 221–225. MR 609655, DOI 10.1090/S0002-9939-1981-0609655-8
- Josef Szabados and A. K. Varma, Inequalities for derivatives of polynomials having real zeros, Approximation theory, III (Proc. Conf., Univ. Texas, Austin, Tex., 1980), Academic Press, New York-London, 1980, pp. 881–887. MR 602815
- József Szabados, Bernstein and Markov type estimates for the derivative of a polynomial with real zeros, Functional analysis and approximation (Oberwolfach, 1980) Internat. Ser. Numer. Math., vol. 60, Birkhäuser, Basel-Boston, Mass., 1981, pp. 177–188. MR 650274
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 93 (1985), 43-47
- MSC: Primary 41A17
- DOI: https://doi.org/10.1090/S0002-9939-1985-0766524-4
- MathSciNet review: 766524