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Some inequalities of algebraic polynomials with nonnegative coefficients


Author: Weiyu Chen
Journal: Trans. Amer. Math. Soc. 347 (1995), 2161-2167
MSC: Primary 41A17
DOI: https://doi.org/10.1090/S0002-9947-1995-1273483-8
MathSciNet review: 1273483
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Abstract: Let $ {S_n}$ be the collection of all algebraic polynomials of degree $ \leqslant n$ with nonnegative coefficients. In this paper we discuss the extremal problem

$\displaystyle \mathop {\sup }\limits_{{p_n}(x) \in {S_n}} \frac{{\int\limits_a^b {{{({{p'}_n}(x))}^2}\omega (x)dx} }} {{\int\limits_a^b {p_n^2(x)\omega (x)dx} }}$

where $ \omega (x)$ is a positive and integrable function. This problem is solved completely in the cases

$\displaystyle ({\text{i}})[a,b] = [ - 1,1],\omega (x) = {(1 - {x^2})^\alpha },\alpha > - 1;$

$\displaystyle ({\text{ii}})[a,b) = [0,\infty ),\omega (x) = {x^\alpha }{e^{ - x}},\alpha > - 1;$

$\displaystyle ({\text{iii}})(a,b) = ( - \infty ,\infty ),\omega (x) = {e^{ - \alpha {x^2}}},\alpha > 0.$

The second case was solved by Varma for some values of $ \alpha $ and by Milovanović completely. We provide a new proof here in this case.

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

  • [1] P. Dörfler, New inequalities of Markov type, SIAM J. Math. Anal. 18 (1987), 490-494. MR 876288 (88f:41024)
  • [2] -, An extremal problem concerning a Markov-type inequality, SIAM J. Math. Anal. 22 (1991), 792-795. MR 1091682 (91k:41024)
  • [3] G. G. Lorentz, Degree of approximation by polynomials with positive coefficients, Math. Ann. 151 (1963), 239-251. MR 0155135 (27:5075)
  • [4] G. V. Milovanović, An extremal problem for polynomials with nonnegative coefficients, Proc. Amer. Math. Soc. 94 (1985), 423-426. MR 787886 (86g:26020)
  • [5] G. V. Milovanović and M. S. Petković, Extremal problems for Lorentz classes of nonnegative polynomials in $ {L^2}$ metric with Jacobi weight, Proc. Amer. Math. Soc. 102 (1988), 283-289. MR 920987 (88k:26016)
  • [6] L. Mirsky, An inequality of the Markov-Bernstein type for polynomials, SIAM J. Math. Anal. 14 (1983), 1004-1008. MR 711180 (84h:41030)
  • [7] J. T. Scheick, Inequalities for derivatives of polynomials of special type, J. Approx. Theory 6 (1972), 354-358. MR 0342909 (49:7653)
  • [8] P. Turán, Remark on a theorem of Erhard Schmidt, Mathematica 2 (1960), 373-378. MR 0132963 (24:A2799)
  • [9] A. K. Varma, Some inequalities of algebraic polynomials having real zeros, Proc. Amer. Math. Soc. 75 (1979), 243-250. MR 532144 (80k:28019)
  • [10] -, Derivatives of polynomials with positive coefficients, Proc. Amer. Math. Soc. 83 (1981), 107-112. MR 619993 (82j:26014)
  • [11] -, Some inequalities of algebraic polynomials, preprint.

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1995-1273483-8
Keywords: Markov inequality, nonnegative coefficients
Article copyright: © Copyright 1995 American Mathematical Society

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