Further improvements of Askey-Steinig’s inequalities for finite sums involving sine and cosine

Alzer, Horst; Kwong, Man

doi:10.1090/proc/15337

Further improvements of Askey-Steinig’s inequalities for finite sums involving sine and cosine
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Proc. Amer. Math. Soc. 149 (2021), 2057-2065 Request permission

Abstract:

In 1974, Askey and Steinig proved that for all $n\geq 0$ and $x\in (0,2\pi )$ the trigonometric sums \begin{equation*} \frac {\sin (x/4)}{1}+\frac {\sin (5x/4)}{2}+\cdots + \frac {\sin ((4n+1)x/4)}{n+1} \end{equation*} and \begin{equation*} \frac {\cos (x/4)}{1}+\frac {\cos (5x/4)}{2}+\cdots + \frac {\cos ((4n+1)x/4)}{n+1} \end{equation*} are positive. Recently, the Askey-Steinig inequalities were improved by the present authors. In this paper, we further improve these inequalities and provide new sharp upper and lower bounds for the two sums given above.

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