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Volume 6, Issue 2, Article 28 |
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On the Inequality of P. Turan for Legendre Polynomials
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Authors: |
Eugen Constantinescu, |
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Keywords:
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Orthogonal polynomials, Legendre polynomials, Turan Inequality, Positivity. |
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Date Received:
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10/01/05 |
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Date Accepted:
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03/02/05 |
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Subject Codes: |
33C10, 26D20.
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Editors: |
Alexandru Lupas (1942-2007), |
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Abstract: |
Our aim is to prove the inequalities where  and  are the Legendre polynomials . At the same time, it is shown that the sequence having as general term is non-decreasing for ![$ xin lbrack -1,1].$](images/027_05_JIPAM/img5.gif)
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![begin{displaymath} frac{1-x^{2}}{n(n+1)}h_{n}leq begin{array}{vert llvert}... ...{1-x^{2}}{2};,quad forall xin lbrack -1,1],;n=1,2,dots, end{displaymath}](images/027_05_JIPAM/img1.gif){kind=small}
![begin{displaymath}n(n+1) begin{array}{vert llvert} P_{n}(x) & P_{n+1}(x) [3pt] P_{n-1}(x) & P_{n}(x) end{array}end{displaymath}](images/027_05_JIPAM/img4.gif)
