论文标题
Chebyshev多项式的一些不平等
Some inequalities for Chebyshev polynomials
论文作者
论文摘要
Askey和Gasper（1976）证明了三角不平等，改善了M. S. Robertson（1945）发现的另一种三角不平等。在这里，这些不平等是根据第一类$ t_n $的Chebyshev多项式重新制定的，然后放入一个不平等的家庭中。发现这些不平等现象的参数的极端值。作为朝着此结果证明的一步，我们确定了以下补充，以$ t_n^{\ prime} $：$ t_n^{\ prime} $：$ t_n^{\ prime}（1）-t_n^{\ prime} {\ prime}（x）\ geq（x）\ geq（1-x） [0,1] \，。 $$通过已知的扩展公式，该属性已扩展到超球级多项式类别$ p_n^{（λ）} $，$λ\ geq 1 $。
Askey and Gasper (1976) proved a trigonometric inequality which improves another trigonometric inequality found by M. S. Robertson (1945). Here these inequalities are reformulated in terms of the Chebyshev polynomial of the first kind $T_n$ and then put into a one-parametric family of inequalities. The extreme value of the parameter is found for which these inequalities hold true. As a step towards the proof of this result we establish the following complement to the finite increment theorem specialized to $T_n^{\prime}$: $$ 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 $P_n^{(λ)}$, $λ\geq 1$.
