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An inequality of Turán type for Jacobi polynomials


Author: George Gasper
Journal: Proc. Amer. Math. Soc. 32 (1972), 435-439
MSC: Primary 33.40
DOI: https://doi.org/10.1090/S0002-9939-1972-0289826-8
MathSciNet review: 0289826
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Abstract: For Jacobi polynomials $ P_n^{(\alpha ,\beta )}(x),\alpha ,\beta > - 1$, let

$\displaystyle {R_n}(x) = \frac{{P_n^{(\alpha ,\beta )}(x)}}{{P_n^{(\alpha ,\beta )}(1)}},\quad {\Delta _n}(x) = R_n^2(x) - {R_{n - 1}}(x){R_{n + 1}}(x).$

We prove that

$\displaystyle {\Delta _n}(x) \geqq \frac{{(\beta - \alpha )(1 - x)}}{{2(n + \alpha + 1)(n + \beta )}}R_n^2(x),\quad - 1 \leqq x \leqq 1,n \geqq 1,$

with equality only for $ x = \pm 1$. This shows that the Turán inequality $ {\Delta _n}(\alpha ) \geqq 0, - 1 \leqq x \leqq 1$, holds if and only if $ \beta \geqq \alpha > - 1$.

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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0289826-8
Keywords: Inequality, Turán inequality, Jacobi polynomials, ultra-spherical polynomials, orthogonal polynomials, Dirichlet kernel
Article copyright: © Copyright 1972 American Mathematical Society

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