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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Some inequalities of algebraic polynomials with nonnegative coefficients


Author: Weiyu Chen
Journal: Trans. Amer. Math. Soc. 347 (1995), 2161-2167
MSC: Primary 41A17
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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1273483-8
PII: S 0002-9947(1995)1273483-8
Keywords: Markov inequality, nonnegative coefficients
Article copyright: © Copyright 1995 American Mathematical Society