  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 $P_n^{(\lambda)}$, $\lambda\geq 1$.
