Some inequalities of algebraic polynomials with nonnegative coefficients

Chen, Weiyu

doi:10.1090/S0002-9947-1995-1273483-8

Some inequalities of algebraic polynomials with nonnegative coefficients
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by Weiyu Chen

Trans. Amer. Math. Soc. 347 (1995), 2161-2167

DOI: https://doi.org/10.1090/S0002-9947-1995-1273483-8

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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 \[ \sup \limits _{{p_n}(x) \in {S_n}} \frac {{\int \limits _a^b {{{({{p’}_n}(x))}^2}\omega (x)dx} }} {{\int \limits _a^b {p_n^2(x)\omega (x)dx} }}\] 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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Some inequalities of algebraic polynomials