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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Some inequalities of algebraic polynomials with nonnegative coefficients
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by Weiyu Chen PDF
Trans. Amer. Math. Soc. 347 (1995), 2161-2167 Request permission

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 \[ \sup \limits _{{p_n}(x) \in {S_n}} \frac {{\int \limits _a^b {{{({{p’}_n}(x))}^2}\omega (x)dx} }} {{\int \limits _a^b {p_n^2(x)\omega (x)dx} }}\] where $\omega (x)$ is a positive and integrable function. This problem is solved completely in the cases \[ ({\text {i}})[a,b] = [ - 1,1],\omega (x) = {(1 - {x^2})^\alpha },\alpha > - 1;\] \[ ({\text {ii}})[a,b) = [0,\infty ),\omega (x) = {x^\alpha }{e^{ - x}},\alpha > - 1;\] \[ ({\text {iii}})(a,b) = ( - \infty ,\infty ),\omega (x) = {e^{ - \alpha {x^2}}},\alpha > 0.\] 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.
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Additional Information
  • © Copyright 1995 American Mathematical Society
  • 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