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Bernstein-type inequalities for the derivatives of constrained polynomials

Author: Tamás Erdélyi
Journal: Proc. Amer. Math. Soc. 112 (1991), 829-838
MSC: Primary 41A17
MathSciNet review: 1036985
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Abstract: Generalizing a number of earlier results, P. Borwein established a sharp Markov-type inequality on $ [ - 1,1]$ for the derivatives of polynomials $ p \in {\pi _n}$ having at most $ k(0 \leq k \leq n)$ zeros in the complex unit disk. Using Lorentz representation and a Markov-type inequality for the derivative of Müntz polynomials due to D. Newman, we give a surprisingly short proof of Borwein's Theorem. The new result of this paper is to obtain a sharp Bernstein-type analogue of Borwein's Theorem. By the same method we prove a sharp Bernstein-type inequality for another wide family of classes of constrained polynomials.

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

  • [1] P. Borwein, Markov's inequality for polynomials with real zeros, Proc. Amer. Math. Soc. 93 (1985), 43-47. MR 766524 (86g:41024)
  • [2] T. Erdélyi, Markov-type estimates for certain classes of constrained polynomials, Constr. Approx. 5 (1989), 347-356. MR 996935 (90f:41009)
  • [3] -, Pointwise estimates for derivatives of polynomials with restricted zeros, Colloq. Math. Soc. J. Bolyai 49; Alfred Haar Memorial Conference (Budapest, 1985), North-Holland, Amsterdam and Budapest, 1987, pp. 329-343. MR 899542 (88f:41025)
  • [4] T. Erdélyi and J. Szabados, Bernstein-type inequalities for a class of polynomials, Acta. Math. Hung. 53 (1989), 237-251. MR 987055 (90f:42001)
  • [5] P. Erdös, On extremal properties of the derivatives of polynomials, Ann. of Math. 41 (1940), 310-313. MR 0001945 (1:323g)
  • [6] G. G. Lorentz, Degree of approximation by polynomials with positive coefficients, Math. Ann. 151 (1963), 239-251. MR 0155135 (27:5075)
  • [7] A. Máté, Inequalities for derivatives of polynomials with restricted zeros, Proc. Amer. Math. Soc. 88 (1981), 221-225. MR 609655 (83g:26022)
  • [8] D. J. Newman, Derivative bounds for Müntz polynomials, J. Approx. Theory 18 (1976), 360-362. MR 0430604 (55:3609)
  • [9] J. T. Scheick, Inequalities for derivatives of polynomials of special type, J. Approx. Theory 6 (1972), 354-358. MR 0342909 (49:7653)
  • [10] J. Szabados, Bernstein and Markov type estimates for the derivative of a polynomial with real zeros, Functional Analysis and Approximation, Birkhäuser Verlag, Basel, 1981, pp. 177-188. MR 650274 (83k:41014)
  • [11] J. Szabados and A. K. Varma, Inequalities for derivatives of polynomials having real zeros, Approximation Theory III (E. W. Cheney, ed.), Academic Press, New York, 1980, pp. 881-888. MR 602815 (82b:26017)

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Keywords: Markov and Bernstein type inequalities, polynomials with restricted zeros
Article copyright: © Copyright 1991 American Mathematical Society

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