Some inequalities for Chebyshev polynomials

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 $T_n^{\prime }$: $$T_n^{\prime }(1)-T_n^{\prime }(x)\ge (1-x)T_n^{\prime \prime }(x),x\in [0,1].$$ By a known expansion formula, this property is extended for the class of ultraspherical polynomials $P_n^{(\lambda )}$, $\lambda \ge 1$.

11 pages