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arXiv · 2001.07013

Some inequalities for Chebyshev polynomials

Geno Nikolov
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Abstract

Askey and Gasper (1976) proved a trigonometric inequality which improves another trigonometric inequality found by M. S. Robertson (1945). Here these inequalities are reformulated in terms of the Chebyshev polynomial of the first kind $T_n$ and then put into a one-parametric family of inequalities. The extreme value of the parameter is found for which these inequalities hold true. As a step towards the proof of this result we establish the following complement to the finite increment theorem specialized to $T_n^{\prime}$: $$ T_n^{\prime}(1)-T_n^{\prime}(x)\geq (1-x)\,T_n^{\prime\prime}(x)\,,\qquad x\in [0,1]\,. $$ By a known expansion formula, this property is extended for the class of ultraspherical polynomials $P_n^{(\lambda)}$, $\lambda\geq 1$.

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Geno Nikolov. 2020-01-20. Some inequalities for Chebyshev polynomials. https://arxiv.org/abs/2001.07013

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