Year: 2020
Type: preprint
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 $P_n^{(λ)}$, $λ\geq 1$.
Source: arXiv (Cornell University)
Author: Nikolov, Geno
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W3002934833