Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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?)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 41A17

Retrieve articles in all journals with MSC: 41A17

Additional Information

Keywords: Markov inequality, nonnegative coefficients
Article copyright: © Copyright 1995 American Mathematical Society

American Mathematical Society