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Inequalities for derivatives of polynomials with restricted zeros


Author: Attila Máté
Journal: Proc. Amer. Math. Soc. 82 (1981), 221-225
MSC: Primary 26C05; Secondary 26D05, 30C10
DOI: https://doi.org/10.1090/S0002-9939-1981-0609655-8
MathSciNet review: 609655
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Abstract: Let $ f$ be a polynomial of degree $ n$ that has real roots all of which lie outside the interval $ ( - 1,1)$. P. Erdös proved that if $ \left\vert f \right\vert \leqslant 1$ on this interval then $ \left\vert {f'} \right\vert < ne/2$ on $ [ - 1,1]$; this is much stronger than the results the inequalities of A. Markov and S. N. Bernstein would give. We will show that if only $ n - k$ roots of $ f$ are restricted as above, then $ \left\vert {f'} \right\vert < {c_k}n$ holds on $ [ - 1,1]$ with appropriate $ {c_k}$. An upper estimate for the best $ {c_k}$ is given. Results for higher derivatives and $ {L^p}$ spaces are also obtained.


References [Enhancements On Off] (What's this?)

  • [1] P. Erdös, On extremal properties of the derivatives of polynomials, Ann. of Math. (2) 41 (1940), 310-313. MR 0001945 (1:323g)
  • [2] W. Feller, An introduction to the theory of probability and its applications, 3rd ed., Vol. I, Wiley, New York, 1968. MR 0228020 (37:3604)
  • [3] A. Máté and P. G. Nevai, Bernstein's inequality in $ {L^p}$ for $ 0 < p < 1$ and $ (C,1)$ bounds for orthogonal polynomials, Ann. of Math. (2) 111 (1980), 145-154. MR 558399 (81c:42003)
  • [4] D. J. Newman, Rational approximation to $ \left\vert x \right\vert$, Michigan Math. J. 11 (1964), 11-14. MR 0171113 (30:1344)
  • [5] A. C. Schaeffer, Inequalities of A. Markoff and S. Bernstein for polynomials and related functions, Bull. Amer. Math. Soc. 47 (1941), 565-579. MR 0005163 (3:111a)
  • [6] J. Szabados and A. K. Varma, Inequalities for derivatives of polynomials having real zeros (manuscript).
  • [7] A. Zygmund, Trigonometric series, Vols. I and II, 2nd ed., Cambridge Univ. Press, Cambridge, 1977. MR 0617944 (58:29731)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1981-0609655-8
Keywords: Bernstein's inequality, Markov's inequality, polynomials with real zeros
Article copyright: © Copyright 1981 American Mathematical Society

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