On a trigonometric inequality of Askey and Steinig

Abstract

In 1974, Askey and Steinig showed that for $\mathit{n}⩾0$ and $\mathit{x}\in \mathrm{(}0\mathrm{,}2\mathit{\pi }\mathrm{)}$ , $$\begin{array}{c}{\mathit{S}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{)}={\displaystyle \sum }_{\mathit{k}=0}^{\mathit{n}}{\displaystyle \frac{sin\mathrm{(}\mathrm{(}\mathit{k}+1\mathrm{/}4\mathrm{)}\mathit{x}\mathrm{)}}{\mathit{k}+1}}\mathrm{>}0\text{and}{\mathit{C}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{)}={\displaystyle \sum }_{\mathit{k}=0}^{\mathit{n}}{\displaystyle \frac{cos\mathrm{(}\mathrm{(}\mathit{k}+1\mathrm{/}4\mathrm{)}\mathit{x}\mathrm{)}}{\mathit{k}+1}}\mathrm{>}0.\end{array}$$ We prove that $$\begin{array}{}\text{(0.1)}& {\mathit{S}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{)}+{\mathit{C}}_{\mathit{n}}\mathrm{(}\mathit{x}\mathrm{)}⩾{\displaystyle \frac{1}{\sqrt{2}}}\end{array}$$ and that the alternating sums $$\begin{array}{c}{\mathit{S}}_{\mathit{n}}^{\ast }\mathrm{(}\mathit{x}\mathrm{)}={\displaystyle \sum }_{\mathit{k}=0}^{\mathit{n}}{\mathrm{(}-1\mathrm{)}}^{\mathit{k}}{\displaystyle \frac{sin\mathrm{(}\mathrm{(}\mathit{k}+1\mathrm{/}4\mathrm{)}\mathit{x}\mathrm{)}}{\mathit{k}+1}}\text{and}{\mathit{C}}_{\mathit{n}}^{\ast }\mathrm{(}\mathit{x}\mathrm{)}={\displaystyle \sum }_{\mathit{k}=0}^{\mathit{n}}{\mathrm{(}-1\mathrm{)}}^{\mathit{k}}{\displaystyle \frac{cos\mathrm{(}\mathrm{(}\mathit{k}+1\mathrm{/}4\mathrm{)}\mathit{x}\mathrm{)}}{\mathit{k}+1}}\end{array}$$ satisfy $$\begin{array}{}\text{(0.2)}& {\mathit{S}}_{\mathit{n}}^{\ast }\mathrm{(}\mathit{x}\mathrm{)}+{\mathit{C}}_{\mathit{n}}^{\ast }\mathrm{(}\mathit{x}\mathrm{)}⩾{\displaystyle \frac{1}{200}}\mathrm{(}13-\sqrt{85}\mathrm{)}\sqrt{300+20\sqrt{85}}=0.41601\dots .\end{array}$$ Both inequalities hold for all $\mathit{n}⩾0$ and $\mathit{x}\in [0\mathrm{,}2\mathit{\pi }]$ . The constant lower bounds given in (0.1) and (0.2) are best possible. The asymptotic behaviour of both sums is also investigated.